Thesis Paulo Lyra

Department of Civil Engineering

University College of Swansea

UNSTRUCTURED GRID ADAPTIVE

ALGORITHMS FOR FLUID DYNAMICS

AND HEAT CONDUCTION

Paulo Roberto Maciel LyraEng.Civil (UFPE), M.Sc. (COPPE/UFRJ)

Thesis submitted to the University of Wales in candidature

for the degree of Doctor of Philosophy

C/Ph/182/94 October 1994

Declaration

This work has not previously been accepted in substance for any

degree and is not being concurrently submitted in candidature forany degree.

Candidate:

Date:

Statement 1

This thesis is the result of my own investigations, except whereotherwise stated. Other sources are acknowledged by footnotes

giving explicit references. A bibliography is appended.

Candidate:

Supervisor:

Date:

Statement 2

I hereby consent for my thesis, if accepted, to be available for

photocopying and for inter-library loan, and for the title and sum-mary to be made available to outside organisations.

Candidate:

Date:

To my parents Francisco Decio and Lilaand my wife Silvana Maria

Acknowledgements

It is really impossible to acknowledge everyone who directly or indirectlycontributed to the realization of this thesis, but I must thank several peoplein particular:

I wish to thank Prof. K. Morgan, my supervisor, for his encouragement,advice and support throughout the course of this work.

Many thanks to my colleagues and collaborators Dr. O. Hassan, F.P. Lin,M.T. Manzari and Dr. P.A.B. de Sampaio for some stimulating work anduseful discussions over the years of my stay in Swansea. I also would like toextend special thanks to Dr. J. Peraire and Dr. N. Weatherill for some helpfuldiscussions.

Thanks to Dr. E.A. de S. Neto for his help with LATEX, including thelayout of this thesis. Thanks to Dr. R.B. Willmersdorf for his help in startingme with UNIX, and for the installation of public domain software used toproduce many drawings and graphics presented in this thesis.

Many thanks to the Department of Civil Engineering as a whole, whichprovided a stimulating environment to pursue this work. Special thanks toMrs. J. Davies and Mrs. L. Newberry for their friendly cooperation.

It is a pleasure to acknowledge my former lecturer Dr. Silvio Romero F.F.Lima for his friendship and continuous interest in my academic life.

Obviously, it would not have been possible to produce this work withoutthe aid of the various books and publications referred to in this thesis, but Imade particular use of the excellent book “Numerical Computation of Internaland External Flows” by Prof. C. Hirsch.

I would like to acknowledge the financial support provided by the BrazilianResearch Council (CNPq – Conselho National de Desenvolvimento Cientıficoe Tecnologico) , under grant No. 204506–90.5, without which all this surelywould have been impossible.

vii

Truly unbounded thanks are due to Silvana, my wife and eternal col-league, who with her love, companionship and patience supported me duringthe realization of this work.

I am much indebted to my family and old friends in Brasil for their un-selfish and permanent support, incentive and love all through my life.

Thanks to those many friends, colleagues, and acquaintances who madethe environment here in Swansea both stimulating and pleasant. A particulargratitude goes to the Brazilian and Friday evening friends who made my stay inthe U.K. more than just an academic experience. It would have been impossiblewithout their close friendship.

Summary

This work is concerned with the development of reliable and versatile com-putational tools for the numerical simulation of two–dimensional heat conduc-tion and incompressible and compressible laminar fluid flow problems. Issuesrelated to adaptive techniques, discretisation methodologies (upwind or cen-tered type) and the design of high–resolution shock–capturing schemes areinvestigated in this thesis.

Three distinct research works have been pursued here. In the first work,attention is focused on the construction of an adaptive finite element pro-cedure with mesh refinement, by mesh enrichment, in time and space, andwith automatic time stepping for the heat conduction problem in a stationarymedium. The Galerkin finite element method and the Euler–backward timemarching scheme are used as the basis to obtain the steady–state and tran-sient approximate solutions. Particular emphasis concentrates on the design ofthe adaptive strategy and the combined influence of time and spatial adapta-tion. The second task is concerned with the derivation of adaptive remeshingstrategies for both steady and unsteady solution of the incompressible Navier–Stokes equations in primitive variables. A Petrov–Galerkin formulation, whichautomatically introduces streamline upwinding and allows equal order inter-polation for all variables, combined with either an explicit or implicit timeintegration represents the general discretisation methodologies adopted. Theadaptive redefinition of the mesh, the error estimate and specific features, suchas the presence of singularities on the solution and accumulation of interpola-tion errors inherent to a transient remeshing, are carefully analysed with someremedies proposed to deal with such difficulties. In the final part of the thesis,the most relevant mathematical–physical properties of the first–order hyper-bolic model equations are discussed. The utilisation of upstream or centereddiscretisation and several ways to produce high–resolution schemes to dealwith this class of problems are described and compared for one–dimensionaltest cases. With regard to upwind discretisation techniques, the most popu-lar flux difference splitting, flux vector splitting and some recently proposed

ix

hybrid splitting methodologies are considered. The higher–order extensionis accomplished by means of a switched artificial viscosity approach, a flux–limited approach or a slope–limited approach. These are presented in a unifiedframework. An edge–based Galerkin finite element formulation is adopted asthe basis for the generalization of the essentially one–dimensional upwindingconcepts intrinsic in most of the approaches mentioned previously. The majorfeatures and implementation details of the proposed finite element edge–basedschemes to solve the compressible Euler equations on unstructured triangulargrids are described. Particular emphasis is given to the computational aspectsrelevant to hypersonic simulation, including stability and convergence rate en-hancement techniques and an adaptive mesh refinement strategy. With regardto the full set of compressible Navier–Stokes equations, a central discretisationof the viscous and heat conduction terms is combined with the high–resolutionschemes used to discretise the inviscid Euler equations.

During the course of this research work, a set of representative one–dimensional and two–dimensional test cases are solved in order to provideevidence of the effectiveness and performance of the proposed formulations.Theoretical solutions, experimental results or well–established numerical pre-dictions are used for comparative purpose, whenever they are available.

Contents

1 Introduction 1

1.1 Motivation and General Considerations . . . . . . . . . . . . . . 1

1.2 Main Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Preliminaries and Governing Equations 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Basic Physical–Mathematical Concepts . . . . . . . . . . . . . 15

2.2.1 Mathematical classification of partial differential equations 15

2.2.2 Initial and boundary conditions . . . . . . . . . . . . . . 21

2.3 Comprehensive Numerical Concepts . . . . . . . . . . . . . . . 23

2.3.1 Variational formulation . . . . . . . . . . . . . . . . . . 23

2.3.2 Weighted residual approximation . . . . . . . . . . . . . 24

2.3.3 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.4 Fundamental definitions: consistency, stability, conver-gence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Conservation Laws & Constitutive Equations . . . . . . . . . . 30

2.4.1 Conservation of mass . . . . . . . . . . . . . . . . . . . 30

2.4.2 Conservation of momentum . . . . . . . . . . . . . . . . 31

2.4.3 Conservation of energy . . . . . . . . . . . . . . . . . . 32

2.4.4 Constitutive equations . . . . . . . . . . . . . . . . . . . 33

2.5 Fluid Dynamics and Heat Transfer Equations . . . . . . . . . . 35

2.5.1 The gas dynamics equations . . . . . . . . . . . . . . . . 36

2.5.2 The incompressible fluid flow equations . . . . . . . . . 40

2.5.3 The heat transfer equation . . . . . . . . . . . . . . . . 42

x

CONTENTS xi

2.5.4 Initial and boundary conditions . . . . . . . . . . . . . . 43

2.6 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.1 Mathematical difficulties . . . . . . . . . . . . . . . . . . 46

2.6.2 Numerical difficulties . . . . . . . . . . . . . . . . . . . . 46

3 Adaptive Procedure for Transient Heat Conduction Simula-tion 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Numerical Solution Procedure . . . . . . . . . . . . . . . . . . . 53

3.2.1 Spatial discretisation . . . . . . . . . . . . . . . . . . . . 53

3.2.2 Time discretisation . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 A–posteriori error estimators . . . . . . . . . . . . . . . 57

3.2.4 Adaptive spatial and time control algorithms . . . . . . . 61

3.3 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.1 Model problem I . . . . . . . . . . . . . . . . . . . . . . 68

3.3.2 Model problem II . . . . . . . . . . . . . . . . . . . . . . 70

3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Adaptive Scheme for Transient Incompressible Viscous FlowComputation 83

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Adaptive Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1 Error estimator . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.3 Accumulation of interpolation errors . . . . . . . . . . . 95

4.3.4 The adaptive algorithm for steady and unsteady flows . 100

4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.1 Steady–state leaky–lid driven cavity flow . . . . . . . . . 102

4.4.2 Simulation of Von Karman vortex street behind a circu-lar cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 105


CONTENTS xii

5 1-D Hyperbolic Equations & Upwind Methods 120

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2 Theoretical Background on Hyperbolic PDE . . . . . . . . . . . 121

5.2.1 Characteristic concept . . . . . . . . . . . . . . . . . . . 121

5.2.2 Weak form of a conservation law . . . . . . . . . . . . . 124

5.2.3 Rankine–Hugoniot relations . . . . . . . . . . . . . . . . 125

5.2.4 Entropy condition . . . . . . . . . . . . . . . . . . . . . 126

5.2.5 System of equations . . . . . . . . . . . . . . . . . . . . . 127

5.2.6 The 1–D Euler equations . . . . . . . . . . . . . . . . . . 130

5.3 Preliminaries on Numerical Methods . . . . . . . . . . . . . . . 135

5.3.1 Basic principles of upwind schemes . . . . . . . . . . . . 135

5.3.2 Non–linear scalar equations . . . . . . . . . . . . . . . . 138

5.4 1–D Euler System of Equations . . . . . . . . . . . . . . . . . . 146

5.4.1 Flux difference splitting approach . . . . . . . . . . . . . 147

5.4.2 Flux vector splitting approach . . . . . . . . . . . . . . 152

5.4.3 Other flux splitting schemes . . . . . . . . . . . . . . . . 155

5.5 Compendium of First–Order Upwind Numerical Fluxes . . . . . 160

5.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . 163

5.6.1 Shock tube problem: rarefaction wave with a sonic point 164

5.6.2 Shock tube problem: hypersonic colliding flow . . . . . . 168

5.6.3 Shock tube problem: slowly moving contact discontinuity 172

5.6.4 Shock tube problem: subsonic regime . . . . . . . . . . . 175

6 High–Resolution Shock–Capturing Methods 184

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.2 Initial Considerations . . . . . . . . . . . . . . . . . . . . . . . . 185

6.2.1 Classical shock–capturing schemes . . . . . . . . . . . . . 187

6.2.2 Advanced shock–capturing schemes . . . . . . . . . . . . 188

6.2.3 Non–oscillatory properties: background . . . . . . . . . 188

6.3 Switched Artificial Viscosity Approaches . . . . . . . . . . . . . 193

6.3.1 Scalar equation . . . . . . . . . . . . . . . . . . . . . . . 194

6.3.2 System of Equations . . . . . . . . . . . . . . . . . . . . 196

6.4 Flux–Limited Methods or Algebraic Approaches . . . . . . . . . 199

CONTENTS xiii

6.4.1 Conditions for low–order LED schemes . . . . . . . . . . 201

6.4.2 Conditions for high–order LED schemes . . . . . . . . . 203

6.4.3 The construction of some high–resolution LED schemes 207

6.5 Slope–Limited Methods or Geometric Approaches . . . . . . . . 212

6.5.1 MUSCL formulation . . . . . . . . . . . . . . . . . . . . 213

6.6 Non–Linear Limiters: A Compilation . . . . . . . . . . . . . . . 216

6.7 Time–Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.7.1 Independent time discretisation . . . . . . . . . . . . . . 222

6.7.2 Lax–Wendroff combined space–time discretisation . . . . 224

6.7.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . 225

6.8 Procedures to build high–resolution schemes . . . . . . . . . . . 227

6.9 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.9.1 Choice of variables to be limited . . . . . . . . . . . . . 229

6.9.2 Influence of the time integration . . . . . . . . . . . . . . 230

6.9.3 Comparison of limiters . . . . . . . . . . . . . . . . . . . 231

6.9.4 Entropy fix . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.9.5 High–resolution schemes: comparative study . . . . . . 241


7 Generalization of High–Resolution Algorithms for 2–D Com-pressible Inviscid Flow Simulation on Triangular Grids 257

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.2 A Finite Element Approach . . . . . . . . . . . . . . . . . . . . 258

7.2.1 Approximate variational formulation . . . . . . . . . . . 259

7.2.2 Edge–based data structure . . . . . . . . . . . . . . . . 261

7.2.3 Time discretisation . . . . . . . . . . . . . . . . . . . . . 264

7.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . 265

7.3 High–Resolution Schemes for 2–D Unstructured Discretisations 266

7.3.1 Conditions to ensure LED property . . . . . . . . . . . 267

7.3.2 The Construction of a Local One–dimensional Stencil . . 270

7.3.3 Unstructured grid solution algorithms . . . . . . . . . . . 274

7.4 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 277

7.4.1 Enhancement of stability and convergence rate . . . . . 277

CONTENTS xiv

7.4.2 Improvement of Accuracy . . . . . . . . . . . . . . . . . 279

7.4.3 The computational implementation . . . . . . . . . . . . 280

7.4.4 Further considerations . . . . . . . . . . . . . . . . . . . 281

7.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 283

7.5.1 Shock tube problem . . . . . . . . . . . . . . . . . . . . 283

7.5.2 Oblique shock on a flat plate . . . . . . . . . . . . . . . . 285

7.5.3 Flow past a cylinder . . . . . . . . . . . . . . . . . . . . 289

7.5.4 Shock interaction on a cylinder . . . . . . . . . . . . . . 290


8 Extension for 2–D Compressible Viscous Flow Simulations 312

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2 Viscous Fluxes Computation . . . . . . . . . . . . . . . . . . . . 313

8.2.1 Standard finite element approach . . . . . . . . . . . . . 314

8.2.2 Edge–based finite element approach . . . . . . . . . . . . 315

8.2.3 Finite element mixed formulation . . . . . . . . . . . . . 315

8.3 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 316

8.3.2 Stability of explicit time integration . . . . . . . . . . . . 316

8.3.3 Wall coefficients . . . . . . . . . . . . . . . . . . . . . . . 317

8.3.4 Further considerations . . . . . . . . . . . . . . . . . . . 318

8.4 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . 318

8.4.1 Supersonic flow past a flat plate . . . . . . . . . . . . . . 319

8.4.2 Hypersonic flow over a compression corner . . . . . . . . 321


9 Conclusions 335

9.1 Summary of Achievements . . . . . . . . . . . . . . . . . . . . . 335

9.2 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 339

9.3 Suggestions for Further Research . . . . . . . . . . . . . . . . . 339

A Time step lower limit 344

B Evaluation of | A | · Z 349

CONTENTS xv

C Finite Differences Interpreted Using Characteristic and Poly-nomial Interpolation Theory 352

D Determination of the Edge–Based Discrete Equation 356

General Notation

Throughout this thesis, an attempt has been made to maintain the no-tation as uniform as possible by assigning specific letter styles to the repre-sentation of each type of mathematical entity. The characters adopted followthe conventional nomenclature used by Computational Fluid Dynamics textbooks and are labelled within the main body of the text. The general schemeof notation employed is described below.

Characters. General usage

• Calligraphic bold-face majuscule TTT , WWW, ...: Spaces. Light-face withsubscripts TJ , WJ , ...: The corresponding components.

• Greek bold-face minuscule σ, τ, ...: Second order tensors. Light-face withsubscripts σij , τij, ...: The corresponding components.

• Greek light-face letters Φ, Ψ, ...: Scalars and scalar valued functions.

• Greek light-face majuscule Γ, Ω, ...: Sets, surfaces, bodies.

• Italic bold-face majuscule A, K, ...: Matrices. Light-face with subscriptsAij , Kij , ...: Components of the corresponding matrices.

• Italic bold-face minuscule v, x, ...: Points, vectors. Light-face with sub-scripts vi, xi, ...: Coordinates (components) of the corresponding points(vectors).

• Italic light-face letters F, f, U, u,...: Scalars and scalar valued functions.

• Slanted sans-serif bold-face minuscule aaa, bbb, ...: Second and fourthorder tensors. Light-face with subscripts aij , bijkl, ...: The correspondingcartesian components.

• Special block-bold majuscule C, H, ...: Discrete operators.

Notation xvii

Acronyms

AD Artificial Dissipation

ADI Alternate Direction Implicit technique

AUSM Advection Upstream Splitting Method

BVP Boundary Value Problem

CD Central Difference

CFD Computational Fluid Dynamics

CFL Courant–Friedrichs–Levy condition

CPU Central Processing Unit

CUSP Convective Upwind and Split Pressure scheme

E-scheme Entropy scheme

ENO Essentially Nonoscillatory scheme

FCT Flux Corrector Transport

FDS Flux Difference Splitting

FVS Flux Vector Splitting

G Galerkin method

GMRES Generalized Minimum Residual method

HLLE Harten–Lax–Van Leer–Einfeldt Riemann solver

IBVP Initial Boundary Value Problem

IVP Initial Value Problem

LED Local Extremum Diminishing

LW Lax–Wendroff method

MUSCL Monotonic Upstream–Centered Schemes

for Conservation Laws

ODE Ordinary Differential Equation

PDE Partial Differential Equation

PPM Piecewise Parabolic Method

R-K Runge–Kutta technique

SLIP Symmetric Limited Positive scheme

SUPG Streamline Upwind Petrov–Galerkin method

TVB Total Variation Bounded

TVD Total Variation Diminishing

USLIP Upstream Limited Positive scheme

WPS Wave/Particle Split

Notation xviii

Subscripts

EN Energy norm

i, j, k, l Normally range between 1,2,3 (x, y, z directions)

I,J,... Nodal point index or component of a set

L2 L2 norm

max Maximum

min Minimum

n Normal component

s Spatial component

t Temporal or tangential component

0 Initial condition

∞ Freestream condition or maximum norm

Supercripts

k Iteration level

n Time level

u Exact, prescribed or averaged value for u

u Approximate value for u

+ Upwind component

− Downwind component

⋆ Variationally recovered or non–dimensional variable

Symbols and operations

∆(·) Increment or variation of (·)∇(·) Spatial gradient of (·)a · b Scalar product of vectors

Ω × I Cartesian product of two sets or spaces

|| · || Norm of vectors

O(·) Order of accuracy

Remark Summation on repeated indices is

implied unless otherwise stated.

Chapter 1

Introduction

1.1 Motivation and General Considerations

There are basically three methodologies utilised in the study of fluid dynam-ics, namely theoretical, experimental and numerical approaches. The task offinding closed solutions for the mathematical model, which describes the phys-ical application, proves to be fruitless in most practical applications. However,even when there is little prospect of finding these solutions, the theoreticalanalysis helps us to find some qualitative information about the solution andto develop theories and techniques to obtain approximate solutions and sostill plays a fundamental role during the validation stage of the alternativeexperimental or numerical simulation techniques. The computer simulationrepresents an attractive alternative for experiments that are difficult, danger-ous, or expensive, and represents the possibility for certain analysis impossiblein laboratory. In particular, it is still economically impossible or very restrictto simple problems the utilisation of models free from empiricism. Therefore,the various branches of fluid dynamics complement and aid each other ratherthan eliminate the other alternatives. Finally, the complexity of the problemsof fluid dynamics requires also a continuous research in observation of the fluidproperties in real scale for the validation of empiric parameters obtained usingthe laboratory models.

Experimentation remains extremely important and even unavoidable forsome practical engineering applications, but the trend towards the increasedependence of design upon numerical simulations has been verified since theappearance of digital computers. The continuous decrease in the cost of per-forming computational simulations when compared to experiments, the as-tonishing development of hardware which allows more complex modelling andfaster simulations and the flexibility and advance of numerical methods withextensive successful predictions consolidate the use of computational fluid dy-namics (CFD) into engineering practice. CFD has already made a large impact

Introduction 2

in design in many areas such as aeronautics, meteorology, nuclear industry,petroleum exploitation, etc. A review of the evolution and role of CFD, to-gether with historical background, can be found, for instance, in references[1, 2, 15, 19, 21, 36, 41], and will not be pursued here.

A general idea of the process of numerical analysis, in the process of de-sign, is illustrated in figure 1.1. The high–level of complexity of the numericalmodelling, which can incorporate many operations designed to reduce humanintervention, is apparent in this flow chart. The interaction with theoreticaland experimental branches of fluid dynamics during the pre–processing andpost–processing stages is also evident. Furthermore, it is clear that the suc-cess of the whole process still relies heavily upon physical intuition, heuristicreasoning and trial–and–error procedure.

Despite the relative maturity reached by CFD, in which the basic method-ologies are, and will remain, well established, computer simulation does nothave quite the same status as physical experiments inside the industry. Thedearth of numerical results concerning certain complex practical applicationsand the remaining doubt about the accuracy of the available techniques forsuch problems still persists at present.

To make computer simulations widely accepted and reliable, intricate flowphenomena in complex two–dimensional and three–dimensional configurationsmust be successfully addressed. This includes assessment of the accuracy ofnumerical solution, the simulation of high speed viscous compressible flows,turbulence modelling, the analysis of equilibrium real gases and flows in chem-ical and thermal nonequilibrium, etc. The fact that larger and faster computersare becoming available will not in itself ultimately resolve the difficulties faced,when dealing with these applications. Also, the necessity for a continuous de-velopment of theories and algorithms will never end, as human necessities andcapacity are limitless. The idealization of models and applications with grow-ing in complexity, the requirements on accuracy and the necessity for instanta-neous response, in some instances, will always leave some desirable simulationsbeyond the reach of the available computers. The basic task of the computa-tional fluid dynamicist is then to combine with the physical understanding ofthe system a judgement and sense of compromise between the required levelof accuracy and the degree of sophistication of the chosen model.

Introduction 3

Real World

Physical Problem

Discrete Solution

- space discretisation (mesh definition, data-structure) - equation discretisation (numerical schemes) - solution parameters (control and flow data) - Enhancement techniques (efficiency, robustness) - etc

- geometry - boundary conditions - source terms - etc.

* Representation of the:

* Resolution of Discrete System of Algebraic Equations

Assessment on the Accuracyof the Approximate Solution

Processing Block

Basic numerical simulation

Mathematical Model

- empiric parameters - geometry - constitutive eqs. - boundary conditions - etc.

* Level of Approximation:

* Assumptions on:

- Spatial - Dynamical - Stediness

Pre-Processing Block

Developing system model

Interpretation of the Results

Visualisation, accuracy/reliabilityanalysis and validation study

Design Improvementsand Optimization

RefineAnalysis

Adapt: - mesh - solution parameters - numerical integration - time-step - etc

* Choice of the:

Post-Processing Block

Verification ofthe predictions

Change Physical Problem

Improve Mathematical Model

Governed by Integral/Differential Equations

Figure 1.1: The process of design via CFD.

Introduction 4

As already mentioned, the basic theory and methodologies in CFD haveachieved a highly developed stage and furnish the general framework for thedesign of computer solution algorithms for many complex fluid flow phenom-ena. This is particularly true when referring to structured–based algorithmimplementations.

Structured mesh methodologies have been developed since the early daysof CFD, and still persist, to the current day, as the most widely used numeri-cal tool of the computational fluid dynamicist. The main reason for this arisesfrom the fact that the analyst can choose an appropriate solution methodfrom among the large number of algorithms which are available. These al-gorithms can normally be implemented in a fairly straightforward and com-putationally efficient manner to produce computer codes for multidimensionalanalysis. The concept of directionality, the limited and small bandwidth ofthe resulting Jacobian matrix and the possibility to devise appropriate proce-dures for interchange of information between nested coarse and finer structuredgrids are some of the features effectively exploited by techniques such as ADItechniques, implicit methods based on sparse matrix technology and multigridwhen applied in connection to structured–based algorithms. However, the largeelapsed time necessary to produce structured grids for extremely complex con-figurations, the difficult control on the quality of the elements (cells) and theunstructured type overhead which arises when adaptive meshing techniquesare implemented represent the main relative disadvantages of the structuredapproach.

The possibility of computing geometrically complex designs becomes a re-ality, as the computational power increases, and is changing the trend of usingstructured into unstructured methodologies over the last decade. This factand the evident natural environment to incorporate adaptivity, which may bethe only hope for resolving very small scale flow features in complex domains,stimulates many CFD practitioners to devote much attention to the develop-ment and use of unstructured mesh methodologies. The enthusiasm of theresearch community has not always been shared by the industrial communityas unstructured mesh methodologies face many drawbacks. The first is theincrease in computational time and memory requirements, as more topologicalinformation needs to be stored and accessed. The second shortcoming consistsof much limited and less efficient computational implementation of a numberof solution algorithms and acceleration techniques. Furthermore, most of theavailable algorithms for unstructured mesh generation are not able to gener-ate fully unstructured meshes suitable for the computation of realistic viscousflows. Finally, insufficient viscous flow calculations using unstructured meshesare available to confirm the effectiveness of the approach to address this classof applications.

A lot has already been done towards the solution of these drawbacks, to

Introduction 5

confirm the accuracy and to render acceptable the extra cost which might re-main when using unstructured approaches. Significant achievements in the de-velopment of data structures and searching algorithms, reducing gather–scatteroperations and memory requirements [6, 9, 23, 31, 33]; the development ofmesh generators able to accurately model complex geometries, with high qual-ity meshes suitable for both inviscid and viscous simulation [5, 17, 35, 39, 40];the derivation of different strategies to assess accuracy and to adapt the dis-cretisation [3, 4, 11, 12, 16, 26, 35, 42, 43]; the design of algorithms whichmimic the alternating direction implicit techniques and the design of strategiesto efficiently transfer informations back and forth between unrelated unstruc-tured meshes used with multigrid acceleration procedure [17, 28, 27, 31, 34];the extension of most of the available methodologies to develop high–ordermulti–dimensional algorithms, such as switched artificial viscosity, MUSCLand many flux–limited or hybrid schemes [6, 7, 8, 13, 22, 23, 29, 37, 38] or useof the Generalized Galerkin Methods [10, 14, 15, 18, 20, 24, 25, 30, 32], madeunstructured mesh approaches into truly competitive solution procedures inCFD. Furthermore, the exploitation of the vector nature of the operations andnatural or forced parallelizations involved in a computational implementationof the solution algorithms on the available vector and parallel computers hascontribute to the effectiveness of these computer tools for fluid dynamics de-sign.

Discretisation methods which are based upon integral methods, such asthe finite volume and finite element method, are natural candidates for usewith unstructured meshes. Despite the intensive and successful utilisationof the finite volume method in CFD, the finite element method represents amuch more general discretisation framework. Finite element method encom-passes not only fluid mechanics but applications are found in almost all fields ofphysical sciences, including solid mechanics, thermal problems, bio–mechanics,acoustics, etc, handling successfully problems governed by elliptic, parabolicand, by now, hyperbolic PDEs, with high degree of sophistication especiallyfor elliptic and parabolic equations. The finite element approach has someextra advantages over the finite volume method, such as the ability to incor-porate differential type boundary conditions naturally; a solid mathematicalfoundation which allows in many instances the estimation of the error in theapproximate solution which can be used to indicate the quality of the approxi-mation throughout the computational domain and so for the adaptation of thediscrete model, and as an indication of the reliability of the numerical solu-tion; it also permits different possibilities for adapting the discrete model, forinstance, the introduction and/or elimination of elements or h–method (meshenrichment and remeshing), moving the position of the nodal points of themesh or r–method, increasing the order of the trial functions or p–method(subspace enrichment), or combination of the previous alternatives ( h–p, r–p,r–h, etc).

Introduction 6

Despite the fact that many ideas developed for structured mesh solverscan now be adopted within the unstructured context, after modifications andadjustments, they normally imply more complexity and less efficiency and a lotyet has to be done to remove any scepticism that still remains in the industrialcommunity for certain areas of applications. Throughout this investigation,unstructured grid adaptive finite element methodologies shall be used as thebasic discretisation technique to address fluid flow and heat conduction simu-lations.

1.2 Main Objectives

Over the last decades, the continuously expanding application of numericalmethods in dealing with engineering and scientific problems makes essentialthe permanent development of new algorithms to improve the quality of thepresent feasible solutions and to enable the solution of previously intractableproblems. The success in designing a computational procedure requires, atleast the following properties of the resulting computer code:

Reliability, i.e. the adopted methods and models must rest on a solid math-ematical basis and within reasonable physical approximations in such a waythat the predicted response is known to be within a selected level of accu-racy. This also leads to the necessity of the use of methodologies to assess thesolution accuracy, the concept of hierarchical models and adaptive procedures;

Robustness, i.e. the structure of the approximations must be numerically stableduring the process for the most comprehensive class of applications in the fieldand with as few adjustable parameters as possible;

Efficiency, i.e. the required response must provide sufficient accuracy, at leastcost and within reasonable time, to allow impact on design. This impliesthe utilisation of appropriate data structures, the exploitation of the currentavailable computer configurations, the use of techniques for enhancement ofstability and convergence rate of the process and the incorporation of adaptivestrategies;

Versatility, i.e. the final computational code must be able to deal with complexgeometries and boundary conditions, different types of loads, general domaindiscretisation and must be portable with little, or no, hardware dependence;

The effort involved to accomplish these requirements for the solution offluid dynamics and heat transfer problems is enormous, as many difficultiesarise in the process. The importance of improvements in any of these relatedareas drives most of the present research in computational fluid dynamics.

The different physical phenomena intrinsic in fluid dynamic applications,i.e. convective and diffusive effects, in a wide range of conditions, leads, in gen-eral, to a hybrid type of mathematical models (elliptic–parabolic–hyperbolic).

Introduction 7

The development of multi–purposed computational codes then renders diffi-cult, or even impossible, as the success either in terms of accuracy, robustnessor efficiency is directly connected to the exploitation of the particular charac-teristics of each class of applications.

As referred to previously, the major advantage of finite element unstruc-tured grid methodologies is the possibility for efficiently implementing adaptivemesh strategies. Adaptivity reduces vulnerability to human errors and assuresmore confidence in the numerical results, which is extremely important for highrisk applications. Furthermore, adaptive techniques stand as an ingenious toolwhose use might represent the possibility to perform certain numerical analysiswithin an affordable time to have an impact in industrial design or even be thedifference between being able or not to carry out such an analysis. The mainobjective of the first part of the present work is to develop and examine adap-tive procedures which exploit some of the physical–mathematical nature of theanalysed problems. Some topics, such as error estimate and indication, com-bined influence of adapting space and time discretisations, how to deal withsingularities, interpolation errors and other features inherent to mesh adaptivestrategies, are examined.

On the other hand, the use of structured discretisation techniques es-sentially represents an advantage in terms of the flexibility and efficient im-plementation of all class of solution algorithms and techniques to enhanceconvergence. In this sense, the principal attention of the second part of thiswork concerns the extension of many well established schemes, set in a finitedifference framework, into unstructured type discretisations. A preliminarycomparative study of some of the available flow solution algorithms is per-formed and some strategies required to enable high speed flow simulations arepresented.

To sum up, the ultimate aim of this research work concerns the devel-opment of adaptive algorithms to deal with heat conduction and fluid flowproblems discretised with unstructured grids. Steady–state and transient so-lution of heat conduction and laminar incompressible fluid flows are studied.Steady–state solutions of laminar compressible inviscid and viscous fluid flowsare addressed and particular emphasis is given to the computational aspectsrelevant to the robust and efficient numerical simulation of high–speed aero-dynamics applications. Many elements of the four requirements mentionedpreviously are considered in this research and the computational codes devel-oped here incorporate some of the most recent advances currently in use withunstructured mesh solution methodologies.

Introduction 8

1.3 Outline of the Thesis

The thesis consists of nine chapters which either contain well establishedmethodologies and general information on computational fluid dynamics ordeal specifically with the proposed formulations and with the applications at-tempted here. The order of presentation follows a chronological and didacticcriteria, and subsequent to this introductory chapter the contents of each chap-ter are now briefly described:

Chapter 2 is devoted to a concise presentation of some mathematical–physical preliminaries concepts, a discussion of some fundamental numeri-cal aspects involved in the discretisation methodologies utilised here and thederivation of the governing equations for heat transfer and fluid flow modelling.

In chapter 3, an adaptive finite element strategy with mesh enrichmentand automatic time stepping control is described for the solution of the scalarparabolic equation which governs the heat conduction in a stationary medium.Model transient problems are included to demonstrate the effectiveness of theprocedure.

Chapter 4 addresses the solution of incompressible Navier–Stokes equa-tions in primitive variables by using an adaptive Petrov–Galerkin formulation.The remeshing technique utilised to adapt the linear triangulation and severalremarks concerning the implementation of this adaptive strategy for steadyand unsteady simulation are discussed. Representative test cases are analysedto show the potentialities of the developed procedure.

In chapters 5 and 6, the 1–D model problems are exploited in the deriva-tion of the schemes to solve the compressible inviscid flow problems. A reviewof the basic features of the hyperbolic partial differential equations and the de-scription of a variety of classical and recent designed upstream discretisationtechniques to solve the Euler system of equations are presented in chapter 5,while the development of high–resolution shock–capturing methods is carriedout in detail through the whole of the chapter 6. A comparative study stress-ing the main peculiarities of each formulation is analysed throughout bothchapters, either conceptually or using numerical evidence.

In chapter 7, the attention is focused on the generalisation of the schemesconsidered in chapters 5 and 6 to deal with the two–dimensional compressibleEuler equations. The construction of unstructured mesh finite element algo-rithms and many issues related to the development of practical computationalcodes for simulation of supersonic and hypersonic flow regimes are examined.A set of illustrative numerical applications, exploiting many features of theproposed formulations, are analysed.

In chapter 8, the extension, by means of a central type discretisation ofthe viscous flux terms, of the schemes described in chapter 7 to the 2–D com-pressible Navier–Stokes equations is presented. Some steady–state supersonic

Introduction 9

and hypersonic flow computations are compared with existing theoretical, nu-merical or experimental data.

In Chapter 9, a summary of the major achievements and conclusions aredrawn from these studies, and some areas for future study are outlined.

Finally, the appendices include some additional information helpful forthe understanding of some specific points addressed in this work.

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[2] Jr. ANDERSON, J.D. Hypersonic and High Temperature Gas Dynamics.MacGraw–Hill, 1989.

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[4] I. BABUSKA, O.C. ZIENKIEWICZ, J.P.R. GAGO, and A. OLIVEIRA,editors. Accuracy Estimates and Adaptive Refinements in Finite ElementComputations. John Wiley & Sons, 1986.

[5] T.J. BAKER. Generation of Tetrahedral Meshes Around a CompleteAircraft. In Pineridge Press, editor, Numerical Grid Generation in Com-putational Fluid Dynamics, pages 675–685, 1988.

[6] T.J. BARTH. Numerical Aspects of Computing Viscous High ReynoldsNumber Flows on Unstructured Meshes. Technical Report 91–0721, AIAAPaper, 1991.

[7] T.J. BARTH and D.C. JESPERSON. The Design and Application ofUpwind Schemes on Unstructured Meshes. Technical Report 89–0366,AIAA Paper, 1989.

[8] J.T. BATINA. Implicit Flux–Split Euler Schemes for Unsteady Aero-dynamic Analysis Involving Unstructured Dynamic Meshes. AIAA J.,29:1836–1843, 1991.

[9] J. BONET and J. PERAIRE. An Alternating Digital Tree (ADT) Al-gorithm for 3–d Geometric Searching and Intersection Problems. Int. J.Num. Meth. Engng., 31:1–17, 1990.

10

Introduction 11

[10] A.N. BROOKS and T.J.R. HUGHES. Streamline Upwind Petrov–Galerkin Formulations for Convection Dominated Flows with ParticularEmphasis on the Incompressible Navier–Stokes Equations. Comp. Meth.Appl. Mech. Engng, 32:199–259, 1982.

[11] L. DEMKOWICZ, P.R.B. DEVLOO, and J.T. ODEN. On an h–TypeMesh Refinement Strategy Based on a Minimization of the InterpolationError. Comp. Methods Appl. Mech. Engng., 3:67–89, 1985.

[12] P.R.B. DEVLOO, J.T. ODEN, and T. STROUBOULIS. Implementationof an Adaptive Refinement Technique for the SUPG Agorithm. Comp.Methods Appl. Mech. Engng., 61:339–358, 1987.

[13] K.P. DIMITRIADIS and M.A. LESCHZINER. A Cell–Vertex TVDScheme for Transonic Viscous Flow. In Proc. of 7th International Con-ference on Numerical Methods in Laminar and Turbulent Flow, Stanford,1991.

[14] J. DONEA. A Taylor–Galerkin Method for Convective Transport Prob-lems. Int. J. Num. Meth. Eng., 20:101–119, 1984.

[15] J. DONEA and L. QUARTAPELLE. An Introduction to Finite ElemetMethod for Transient Advection Problems. Comp. Meth. Appl. Mech.Engng., 95:169–203, 1992.

[16] J.P.R. GAGO, D.W. KELLY, O.C. ZIENKIEWICZ, and I. BABUSKA.A Posteriori Error Analysis and Adaptive Processes in the Finite ElementMethod – part 1. Int. J. Num. Meth. Engng., 19:1621–1656, 1983.

[17] O. HASSAN, E.J. PROBERT, K. MORGAN, and J. PERAIRE. LineRelaxation Methods for the Solution of 2D and 3D Compressible ViscousFlows Using Unstructured Meshes. In Proc. “Recent Developments andApplications in Aeronautical CFD”, Bristol, 1993.

[18] J.C. HEINRICH, P.S. HUYAKORN, O.C. ZIENKIEWICZ, and A.R.MITCHELL. An Upwind Finite Element Scheme for Two–DimensionalConvective Transport Equations. Int. J. Num. Meth. in Engng., 11:131–143, 1977.

[19] C. HIRSCH. Numerical Computation of Internal and External Flows,volume 2. John Wiley & Sons, 1990.

[20] T.J.R. HUGHES, L.P. FRANCA, and M. BALLESTRA. A New FiniteElement Formulation for Computational Fluid Dynamics: V. Circumvent-ing the Babuska–Brezzi Condition: a Stable Petrov–Galerkin Formulationof the Stokes Problem Accomodating Equal–Order Interpolations. Comp.Meth. Appl. Mech. Engng., 59:85–99, 1986.

Introduction 12

[21] A. JAMESON. Success and challenges in computational aerodynamics.Technical Report 87–1184, AIAA Paper, 1987.

[22] A. JAMESON. Artificial Diffusion, Upwind Biasing, Limiters and theirEffect on Accuracy and Multigrid Convergence in Transonic and Hyper-sonic Flows. Technical Report 93–3359, AIAA Paper, 1993.

[23] A. JAMESON, T.J. BAKER, and N.P. WEATHERILL. Calculation ofInviscid Transonic Flow Over a Complete Aircraft. Technical Report 86–0103, AIAA Paper, 1986.

[24] B.N. JIANG and G.F. CAREY. A Stable Least–Squares Finite ElementMethod for Non–Linear Hyperbolic Problems. Int. J. Num. Meth. inFluids, 8:993–942, 1988.

[25] C. JOHNSON. Numerical Solution of Partial Differential Equations bythe Finite Element Method. Cambridge University Press, 1987.

[26] C. JOHNSON, Y.Y. NIE, and V. THOMEE. An a Posteriori Error Esti-mate and Automatic Time Step Control for a Backward Discretization ofa Parabolic Problem. SIAM J. of Numerical Analysis, 27:277–291, 1990.

[27] M.-P. LECLERCQ, J. PERIAUX, and B. STOUFFLET. Multigrid Meth-ods with Unstructured Meshes. In Proceedings of the 7th InternationalConference on Finite Elements in Flow Problems, pages 1113–1118, Hun-stsville, Alabama, 1989.

[28] R. LOHNER and K. MORGAN. Unstructured Multigrid Methods forElliptic Problems. Int. J. Num. Meth. Fluids, 24:101–115, 1986.

[29] R. LOHNER, K. MORGAN, J. PERAIRE, and M. VAHDATI. Finite El-ement Flux–Corrected Transport (FEM–FCT) for the Euler and Navier–Stokes Equations. Int. J. Num. Meth. Fluids, 7:1093–1109, 1987.

[30] R. LOHNER, K. MORGAN, and O.C. ZIENKIEWICZ. The Solution ofNon–Linear Hyperbolic Equation Systems by the Finite Element Method.Int. J. Num. Meth. Fluids, 4:1043–1063, 1984.

[31] D.J. MAVRIPLIS. Multigrid Solution of Two–Dimensional Euler Equa-tions on Unstructured Triangular Meshes. AIAA J., 26:824–831, 1988.

[32] K.W. MORTON. Generalised Galerkin Methods for Steady and UnsteadyProblems. In K.W. Morton and M.J. Baines, editors, Numerical Methodsfor Fluid Dynamics, pages 1–32. Academic Press, 1982.

[33] J. PERAIRE, J. PEIRO, and K. MORGAN. A 3–D Finite Element Multi-grid Solver for the Euler Equations. Technical Report 92–0449, AIAAPaper, 1992.

Introduction 13

[34] J. PERAIRE, J. PEIRO, and K. MORGAN. Finite Element MultigridSolution of Euler Flows Past Installed Aero–Engines. Computational Me-chanics, 11:433–451, 1993.

[35] J. PERAIRE, M. VAHDATI, K. MORGAN, and O.C. ZIENKIEWICZ.Adaptive Remeshing for Compressible Flow Computations. J. Comp.Phys., 72:449–466, 1987.

[36] P.J. ROACH. Computational Fluid Dynamics. Hermosa Publishers, 1976.

[37] S. SOLTANI. An Upwind Scheme for the Equations of Compressible Flowon Unstructured Grids. PhD thesis, University of London – ImperialCollege of Science, Technology and Medicine, 1991.

[38] R.R. THAREJA, R.K. PRABHU, J. MORGAN, K. nad PERAIRE,J. PEIRO, and S. SOLTANI. Applications of an Adaptive UnstructuredSolution Algorithm to the Analysis of High Speed Flows. Technical Report90–0395, AIAA Paper, 1990.

[39] N.P. WEATHERILL. Mesh Generation in Computational Fluid Dynam-ics. Technical Report 1990–10, Von–Karman Institute for Fluid DynamicsLecture Notes, 1990.

[40] M.A. YERRY and M.S. SHEPHARD. Automatic Three DimensionalMesh Generation by Modified–Octree Technique. Int. Num. Meth. En-gng., 20, 1990.

[41] O.C. ZIENKIEWICZ and R.L. TAYLOR. The Finite Element Method:Solid and Fluid Mechanics, Dynamics and Non–linearity, volume 2.Macgraw–Hill, 1991.

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[43] O.C. ZIENKIEWICZ and J.Z. ZHU. Superconvergent Recovery Tech-niques and A Posteriori Error Estimation in F.E.M. – Part 1 A Gen-eral Superconvergent Recovery Technique. Int. J. Num. Meth. Engng.,33:1331–1382, 1992.

Chapter 2

Preliminaries and GoverningEquations

2.1 Introduction

The exact prediction of a physical system frequently cannot be obtained, evenwith the adoption of the most refined mathematical model. Therefore, the keystep in an engineering or scientific analysis is the choice of a suitable mathemat-ical model, the knowledge of its limitations and the physical approximationsassumed in its derivation. From a practical point of view, when a numericalsolution of the model is attempted, the complexity of the adopted model islimited by the computational equipment available and has been expanded withthe evolution of computer technology.

Partial differential equations (PDE) are at the heart of many, if not most,computer analyses or simulations of continuous physical systems, such as fluidmechanics, heat conduction, solid mechanics, etc. As a result, a broad classof methods has been developed to solve PDE, such as separation of variables,integral transform, numerical methods, etc. All of these have their own im-portance and achievements in the solution of PDE, but none of them is asflexible and general as the numerical methods, which allow complex geometryconfigurations, non-linearities and coupled systems of partial differential equa-tions. Numerous approximations are involved in a computational simulationof a fluid mechanics problem, beginning with the domain of continuous me-chanics, passing through numerical techniques and ending up with the analystinterpreting and judging the results. Since the numerical scheme adopted willonly solve the mathematical model, which idealizes the physical problem, theassumptions in this model will be reflected in the predicted response, and noone can expect any other information in the simulation of the physical phe-nomena than the information contained in the mathematical model. Hence,

Preliminaries & Governing Equations 15

the choice of an appropriate mathematical model and the physical assumptionsinvolved in its derivation are crucial and completely decide the insight into theactual physical problem that will be obtained by the analysis. The approxi-mations inherent in numerical simulation will be detailed in the remainder ofthis thesis. Errors in the final analysis of the results can never be completelyavoided since it concerns pure human factors, but the inclusion of self-adaptiveprocedures, optimization techniques and the utilization of visualization toolscan considerably reduce this source of error, and will be partially the themeof the next chapters.

The mathematical idealization of the physical problem normally leads, asalready mentioned, to differential equations and the awareness of some of thefundamental aspects of the mathematical models are important for a broaderunderstanding of the numerical analysis. Such fundamental aspects will beshortly presented in the following section. In section 2.3 the attention is fo-cused on some generic concepts involved in the discretisation techniques to beadopted. In sections 2.4 and 2.5 the mathematical formulation of the fluid dy-namics and heat transfer problems, and many important physical concepts in-volved in their determination, are briefly reviewed. Finally, certain difficultiesinherent in numerical simulations and general considerations are mentioned.

2.2 Basic Physical–Mathematical Concepts

2.2.1 Mathematical classification of partial differentialequations

From a purely mathematical point of view, there are many ways to classify apartial differential equation, and some of the concepts involved in these classi-fications should be recalled as they will be extensively used here. Nevertheless,the straight connection between the mathematical nature of the equation andthe physical phenomena involved represents a more natural way for the classi-fication of partial differential equations studied in fluid mechanics. For morerigorous mathematical definitions of partial differential equations and theirclassification see [1, 9, 10, 13].

Before tackling the complications of coupled systems of equations, it isinteresting to introduce a simplified scalar problem which models the complexsystems of interest. By far the most popular model problem in computationalfluid dynamics is given by the Burgers equation and by the equations deducedfrom it. These models are extensively utilized for comparisons and valida-tion of new numerical methods since a large number of exact solutions areknown [4, 36]. The one–dimensional viscous Burgers equation, together withits linearized version, can be written in a generalised conservative form as


∂U

∂t+

∂F

∂x=

∂G

∂x(2.1)

where:

U = au(x, t) (Unknown)

F = (b + cu

2)u (Convective Flux)

G = d∂u

∂x(Diffusive Flux)

(2.2)

or in a quasi–linear form as

a∂u

∂t+ (b + cu)

∂u

∂x= d

∂2u

∂x2(2.3)

with a,b,c,d being free constant parameters. This expanded Burgers equationpossesses all the most important characteristics present in the equations thatgovern fluid dynamics and heat transfer phenomena. The contribution of eachterm in (2.1), is directly related with the physical behavior of the problem andwill determine the mathematical nature of the partial differential equation.

In table 2.1 some of the most important model equations derived fromequation (2.1), according to the choice of the free parameters are presented,and a mathematical classification with an example of a physical phenomenongoverned by the resulting equation is also presented. In figures 2.1, 2.5, 2.6and 2.7, the initial sinusoidal data and a typical resulting solution for each ofthe model equations presented in table 2.1 is shown. The effects of convec-tion, diffusion and transient terms are clearly observed, and the nature of themathematical equations and its influence in the main features of the solutionis also noticed and will be detailed below.

The parameter a determines if the time dependent term is present or notin equation (2.1). In the first case in which (a 6= 0) the solution is sought duringits evolution in time, depending both on the initial and on the boundary condi-tions. This is referred to, in mathematical terminology, as an initial/boundaryvalue problem (IBVP). In the second case in which (a = 0) the solution isstationary in time and determined by the boundary conditions only. This isreferred to as boundary value problem (BVP). The distinction between thesetwo classes of problems was already well defined by Richardson [25] when hedescribed the first as a marching problem, where the solution marches from onestage to the other guided and modified in its way by the boundary conditions.The second, as a jury problem, since the solution in a given point dependson all surrounding points and boundary conditions. From a computational


Transient Convective Diffusive Equation Mathematical Example

Term Term Term Classification

a 6= 0 b = 0 d 6= 0 Diffusion Parabolic Transient Heatc = 0 Conduction

a = 0 b = 0 d 6= 0 Laplace’s Elliptic Steady Heatc = 0 Conduction

a 6= 0 b 6= 0 d = 0 Convection Hyperbolic Linear Wavec = 0 (Advection) Propagation

a 6= 0 b = 0 d = 0 Inviscid Hyperbolic Non–Linear Wavec 6= 0 Burgers Propagation

Linear Transporta 6= 0 b 6= 0 d 6= 0 Convection Parabolic & Dispersion of

c = 0 Diffusion a Pollutant

Non–Linear Transporta 6= 0 b = 0 d 6= 0 Viscous Parabolic & Dispersion of

c 6= 0 Burgers a Pollutant

Table 2.1: Scalar model problems in CFD.

point of view, this classification is of paramount importance and the goal of anumerical code should be to track the time evolution of the system with somedesired accuracy for the propagation problems and to converge on the propersteady solution everywhere for the equilibrium problems.

(a) Transient (b) Steady

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

--><--

u(x,0)u(x,t)u

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

--><--

u(x,0)u(x,t)u

Figure 2.1: Pure diffusion solutions.

In equation (2.3) the parameter d regulates the amount of diffusion or dis-sipation present in the problem, and appears through a second–order derivative


term. The effect of pure diffusion, i.e. no convection (b = c = 0), described bythe transient heat conduction equation can be observed in figure 2.1(a), wherethe solution propagates in the space direction but is damped in time. Thefinal equilibrium solution is shown in figure 2.1(b), for zero values prescribedas boundary conditions.

The equation (2.3) is of second–order, which refers to the order of thehighest partial derivative in the equation. Moreover, it is quasi–linear, i.e.such that even being in general non–linear it is linear in the highest derivative[1]. The second term in equation (2.3) represents the transport properties ofthe flow and according to the choice of parameters b and c the non–lineareffects can be suppressed (b 6= 0, c = 0), with the parameter b representing theconvective velocity in the resulting linear wave equation.

x

y

PΩ

Γ

Domains ofDependence and Influence

Figure 2.2: Typical domains of dependence and influence of a point P for a2-D elliptic problem.

Another very useful classification, based on the characteristic or curvesof information propagation [1, 13], distinguishes three different categories ofpartial differential equations: elliptic, parabolic and hyperbolic. In the ellipticproblem the solution in any point depends on and influences all its surround-ing points, figure 2.2, which is precisely an equilibrium or boundary valueproblem. For example, in the steady–state heat conduction equation, withoutthe presence of source terms, the temperature at the end of the domain influ-ences the temperature at any point in the domain and when the equilibriumis achieved only the boundary condition dictates the final solution (see figure


2.1(b)). Parabolic and hyperbolic equations are subclasses of marching or ini-tial/boundary value problems, where the fundamental difference is related tothe limit on the domain of dependence and zone of influence of the solution ina given point, see figures 2.3 and 2.4.

P

t

x

T

Ω ΓΓL R

Domain ofInfluence

Domain ofDependence

t0

Figure 2.3: Typical domains of dependence and influence of a point P for a1-D parabolic problem.

P

t

x

T

Ω ΓΓL R

Domain ofInfluence

Domain ofDependence

t0

Figure 2.4: Typical domains of dependence and influence of a point P for a1-D hyperbolic problem with two straight characteristics per point.

In the parabolic problem there are no limited zone of dependence andinfluence with the solution at a given time being dependent and influencing theentire physical domain including the boundary points, e.g. the temperatureat a fixed time and point is dependent on the temperature on all points atthe domain in a previous time and will influences the temperature of the fulldomain in the future. On the other hand, the main property of hyperbolic


equations is the limited domain of dependence, where the information at aspecific point of the domain is only influenced by a specific region defined by thewave–propagation directions, which also determine the zone of influence of thatpoint. An idea of solution behavior for the parabolic heat conduction equationis given in figure 2.1(a) and for the hyperbolic pure convection equation andinviscid Burgers equation is presented in figure 2.5.

(a) Linear (b) Non–Linear

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

-->

u(x,0)u(x,t)u

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

-->

u(x,0)u(x,t)u

Figure 2.5: Pure convection solutions.

In figure 2.5 the pure convection or transport effect for the linear waveequation and non–linear inviscid Burgers equation are shown. The linear equa-tion just convects the wave while the non–linear counterpart changes com-pletely the initial profile which leads to a breakdown of the continuous solutionand the appearance of a shock discontinuity and an expansion fan [14]. Figure2.6 shows the solutions, considering both convective and diffusive terms, forlinear and non-linear cases. It should be observed that no discontinuous solu-tion actually exists due to the presence of the dissipative term which gives aparabolic character for the convection–diffusion and viscous Burgers equations.

Finally, the mixed effect of diffusion and convection is presented in figure2.7 in which the parameter ra represent the ratio between the convective anddiffusive terms, being equal to b/d for the linear equation and (cu)/d for thenon–linear equation. When ra assumes small values the diffusive effect isdominant and the problem behavior is parabolic. However, in the limit ofvanishing of the diffusion term (ra → ∞), i.e. negligible if compared to theconvection term, the pure convection model is recovered and the solution wouldbe nearly discontinuous, in the sense that the change in u would occur over adistance that is microscopic when compared with the natural length scale of theproblem. In such a case, the hyperbolic nature of the equation dominates withthe implications already mentioned. Due to the importance of the hyperbolictype equations when dealing with compressible fluid flow, a detailed description



0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

u(x,0)u(x,t)

u

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6X

u(x,0)u(x,t)u

Figure 2.6: Mixed convection–diffusion solutions.


0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6

u(x,0) ra --> 0 ra --> 100ra --> oo

u

X

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6

u(x,0) ra --> 0 ra --> 100ra --> oo u

X

Figure 2.7: Study of convection–diffusion effects.

of the basic elements of the theory of hyperbolic partial differential equationsand their consequence when devising numerical techniques for their solution ispresented later in this work.

2.2.2 Initial and boundary conditions

The solution of all categories of partial differential equations depends uponthe initial and/or boundary conditions, as discussed in the previous section.Furthermore, the question whether or not the problem is well–posed, is alsorelated to these conditions, with a problem being well–posed, in the senseof Hadamard, if the solution to the problem exist, is unique and dependscontinuously upon the initial and boundary conditions [13]. The proof of thesequalitative informations about the solution for a given problem is a subject of


mathematical analysis [21, 24] and therefore beyond the scope of this work, butsome considerations concerning appropriate initial and boundary conditions fora scalar problem are pertinent.

Ω

ΓL ΓR

Γ U ΓR= ΓL

Figure 2.8: 1-D finite domain sketch.

For the scalar problem described in equation (2.1) and considering thedomain represented in figure 2.8, the initial condition is specified assumingthat

u(x, t0) = u0(x) for all x in Ω at time t = t0 (2.4)

where u0 represents a known distribution of the dependent variable u at timet0, and Ω denotes the finite spatial domain.

From a purely mathematical point of view, there are two basic types ofboundary conditions for a set of second–order partial differential equations

u = u(x, t) on ΓD at t ≥ t0 (2.5)

where u represents a prescribed value of the solution on boundary ΓD and

∂u

∂x= g(u, x, t) on ΓN at t ≥ t0 (2.6)

where g represents a specified value of the derivative of the variable u onboundary ΓN . The first type described, is called Dirichlet or essential bound-ary condition and the second type is referred to as Von Neumann or naturalboundary condition. A third type, represented by a combination of Dirich-let and Neumann on the same boundary is referred to as Robin’s or as amixed boundary condition. For the elliptic and parabolic equations describedin previous section in both boundaries ΓL and ΓR either Dirichlet or Neu-mann boundary conditions must be imposed, with Dirichlet type imposed atleast once in order to obtain a unique distribution of the variable u. However,for the hyperbolic problems presented, only first–order spatial derivatives arepresent and only Dirichlet type boundary condition can be applied. Due tothe one–directional nature of the solution propagation only the boundary withthe incoming information can be prescribed and the other must be left free. Adetailed description of the correct boundary condition for hyperbolic partialdifferential equations will be presented in chapter 5.


2.3 Comprehensive Numerical Concepts

2.3.1 Variational formulation

As closed solutions of the mathematical models are possible only for verysimple cases one has to rely on numerical methods for finding approximatesolutions. Many discrete methods can be interpreted as derived from weightedresidual methods [38], in particular the methods adopted in this work fallwithin this category. To introduce the discrete procedure, let us consider aninitial boundary value problem, such as the problems described by equations(2.1) and (2.4) to (2.6), which can be stated in a general form as: Find asolution for

LΦ − P = 0 in Ω = (x, t)|(x, t) ∈ Ω × IMΦ − Q = 0 on Γ = (x, t)|(x, t) ∈ Γ × IΦ(x, t0) = Φ0(x) for all x ∈ Ω ⊂ RRRn at t = t0

(2.7)

where x represents the spatial coordinates, I is the time interval (t0, T ), L andM are differential operators with order l and m respectively, P and Q areknown functions independent of function Φ, which is the exact solution for theproblem.

The starting point for the construction of a weighted residual approxima-tion for the problem represented in (2.7) is the identification of an appropriatevariational formulation for the problem [15]. This can be achieved by intro-ducing a trial or interpolation function set, TTT , and the weighting function set,WWW, defined such that

TTT = Φ(x, t) ∈ CCCk |Φ = Φ0 on Ω at t = t0WWW = W(x, t)

(2.8)

where CCCk represent the set of all functions k-times continuously differentiablewith respect to x and t, with k = max(l, m), and the weighting functions setsWWW do not need to satisfy any continuity requirements. A possible variationalformulation [20] of the problem given in equation (2.7) is then

find Φ ∈ TTT such that ∀W(x, t) ∈ WWW and t > t0∫

ΩWΩ(LΦ− P ) dΩ +

∫

ΓW Γ(MΦ − Q ) dΓ = 0

(2.9)

where the weighting functions can be chosen independently WΩ and W Γ for theintegrals over Ω and Γ. The fundamental lemma of the calculus of variations


[8] assures that a solution Φ for the equivalent problem (2.9) is identical tothe exact solution of the problem formulated in the classical differential form(2.7).

2.3.2 Weighted residual approximation

To obtain an approximate solution to the problem posed in the variationalform (2.9) we consider a basis T1, T2, . . . for the trial function set and definethe subset TTT (p) of the trial function set TTT which is spanned by the first pbasis functions TI , i.e.

TTT (p) = Φ(x, t) | Φ =p∑

I=1

aITI and Φ = Φ0 on Ω at t = t0 (2.10)

where aI are unknown coefficients. By substituting the approximate solutionΦ (2.10) into equations (2.7) the discrete counterpart of the problem (2.7) canbe written as: Find an approximate solution for

LΦ − P = RΩ

in Ω = (x, t)|(x, t) ∈ Ω × IMΦ − Q = R

Γon Γ = (x, t)|(x, t) ∈ Γ × I

Φ(x, t0) = Φ0(x) for all x ∈ Ω ⊂ RRRn at t = t0

(2.11)

where RΩ

and RΓ

denote the portion of the total residual R = Φ−Φ introduced

in the domain Ω and on the boundary Γ equations by the errors inherent inthe approximation given by (2.10). The residuals R

Ωand R

Γare functions of

(x, t) and an attempt to reduce these residuals (RΩ→ 0 and R

Γ→ 0) should

lead to a good approximation for the required solution Φ(x, t). The weightedresidual method seeks an approximate solution Φ to the discrete approximatevariational formulation, which can be stated according to

find Φ ∈ TTT (p) such that for eachWJ (x, t) ∈ WWW and t > t0∫

ΩWΩ

J (LΦ − P )︸︷︷︸R

Ω

dΩ +∫

ΓW Γ

J (MΦ− Q )︸︷︷︸R

Γ

dΓ = 0 (2.12)

This solution method is equivalent to Rayleigh–Ritz method [38], which looksfor a stationary solution or extrema of a natural variational or functional for-mulation corresponding to the original problem, if such a functional is avail-able. Due to many practical reasons [38], the so called partial discretisation


procedure is normally preferred when dealing with time–dependent problems.In this procedure two steps are necessary; in the first step, the functions TI

in equation (2.10) are chosen to be dependent only on the space coordinatesTI(x) and the coefficients aI are functions of the time coordinate aI(t). Insuch a case , in which aI is no longer a constant, this solution procedure isequivalent to Kantorovic’s method [8], whenever a natural variational principleexist for the particular problem of interest.

The weighted residual statement equivalent to (2.12) is obtained by re-placing R

Ω, R

Γby RΩ and RΓ respectively, and becomes

find Φ ∈ TTT (p) such that for eachWJ (x, t) ∈ WWW and t > t0∫

ΩWΩ

J (LΦ − P )︸︷︷︸

RΩ

dΩ +∫

ΓWΓ

J (MΦ− Q )︸︷︷︸

RΓ

dΓ = 0 (2.13)

By letting J = 1, · · · , p a system of ordinary differential equations forthe unknown parameters aI(t) is generated. This system can be reduced toan algebraic set of equations, in the second step, after repeating the weightedresidual technique with the functions T n(t) depending on the time coordinateand the coefficients an being unknown constants for a certain time T, i.e. byrequiring that the integral of the weighted residual RT , introduced by theapproximated solution for the ordinary differential equations, performed overthe time interval (t0,T), must be zero.

Once more the fundamental lemma of the calculus of variations [8] guar-antee that the solution of the problem stated in equation (2.13) is equivalentto the solution of the problem given in (2.11). The general convergence re-quirement assert that Φ → Φ as p → ∞, i.e. TTT (p) ≡ TTT . This requirement isconvertible to the requirement present in (2.13), i.e. that the integrals of theresiduals must be zero for every WJ in WWW and for all time t > t0 as p → ∞.In this way, if a very large value for p is adopted, in principle, the coefficientsaI(t) can be chosen so that the approximation Φ must approach the exact so-lution Φ. Different forms of weighting functions set WWW lead to different classof weighted residual approximation methods. Some common choices are:

Point Collocation Method

WWW = WJ | WJ = δ(x − xJ) (2.14)

where δ is the Dirac delta function. This procedure is equivalent to simplysetting R(xJ , t) = 0 at p chosen points xJ in Ω. It is interesting to remarkthat the basic finite difference methods can be verified as collocation methodswith locally defined basis functions.


Subdomain Method

WWW = WJ | WJ = 1 ∀ x in ΩJ and WJ = 0 ∀ x outside ΩJ (2.15)

where the domain is divided into p subdomains ΩJ . This method is closelyrelated to the finite volume method, and it makes the integral of the residualover each subdomains ΩJ zero, which provides an appropriate framework forenforcing conservation properties inherent in the governing equations, at thediscretised equation level.

Least–Squares Method

WWW =

WJ | WJ =

∂R

∂aJ

(2.16)

which is equivalent to minimizing the square of the residual R with respect tothe free parameters aJ , i.e.

∂

∂aJ

∫

ΩR2

Ω dΩ +∫

ΓR2

Γ dΓ

= 0 (2.17)

The most important property of the least squares method is the fact that italways generate a symmetric coefficient matrix, in the final algebraic systemof equations, independent of the properties of the operators L and M.

Galerkin Method

WWW = WJ | WJ = TJ (2.18)

it is also called Bubnov–Galerkin method and simply takes the weighting func-tions to be the same as the trial functions. The implication of this choice [3]is the fact that it leads to symmetric ( and also positive definite) coefficientmatrix if the operators L and M are symmetric (and also positive definite).In addition, it leads to the best approximate solution in the energy norm[15]. These reasons make the Galerkin method the most popular choice andit is the basis for the standard finite element method. It is worthwhile tomention that, in the literature, all weighted residual approximation methodswhich adopt weighting functions from a set which is different to the set of trialfunctions are referred to as Petrov-Galerkin methods.


2.3.3 Weak solutions

Considering a particular set of weighting function WWW such that its elements WJ

have appropriate level of continuity, an alternative, so called weak, formulationto (2.13) is given by

find Φ ∈ TTT (p) such that for eachWJ(x, t) ∈ WWW and t > t0∫

Ω( CWΩ

J )(DΦ ) dΩ +∫

ΓWΩ

J ( EΦ ) dΓ +∫

ΓWΓ

J (MΦ − Q ) dΓ = 0

(2.19)

where C, D and E are differential operators involving an order of differentia-tion lower than the original operator L. If Φ satisfies (2.19) for every suitableweighting functions WJ then Φ is called a weak solution of (2.7), and a broaderclass of approximated solutions Φ is possible, since less constraint is required forthe choice of the trial function set TTT . With a suitable choice of the boundaryweighting functions WΓ

J , it may be possible to cancel the second term in equa-tion (2.19) with part of the last term over the boundary ΓN . The residual RΓ

on the portion of the boundary ΓD with Dirichlet type boundary condition isidentically null and so not automatically imposed in the variational statement.Therefore, trial solutions, based upon the solution of a variational formula-tion, must explicitly satisfy the Dirichlet type boundary condition, which isalso called essential boundary condition. On the other hand, the remain por-tion of the boundary have their boundary conditions fulfilled implicitly by thesatisfaction of the variational statement. In this case, the boundary conditionsare called natural boundary condition for the variational formulation [39].

When the finite element method is applied to solve at most second–orderpartial differential equations, we usually adopt an approximate weak varia-tional formulation, such as given in equation (2.19), obtained from (2.13) bya single integration by parts of the higher–order differential terms. However,instead of use the weighted residual statement on Ω, the domain Ω is first sub-divided into nonoverlapping subdomains or elements ΩE , connected throughnodal points, with the approximate solution written directly in terms of thenodal unknowns (aI = ΦI) and the trial functions being in general chosen fromlower–order piecewise polynomials restricted to contiguous elements.

2.3.4 Fundamental definitions: consistency, stability, con-vergence

The objective here is to give a short review of some important concepts requiredfor numerical schemes developed to solve partial differential equations. Moredetailed and formal discussion can be found in the literature [1, 13, 31].


Initially, it is necessary to define the truncation error εT of a given dis-crete equation, which represents the difference between the partial differentialequation and the corresponding discrete equation. This allows the definitionof consistency, which states that the discrete equation should tend to the cor-responding continuous equation under refinement of the spatial and time dis-cretisation , i.e.

lim∆x,∆t→0

εT = 0 (2.20)

The utilization of Taylor series allows the verification of consistency for a givenscheme and the determination of the order of accuracy, or the rate at whichthe discrete equation tends to the differential equation as ∆x, ∆t tend to zero.The truncation error is generically expressed in the form

εT = O(∆xr, ∆ts) (2.21)

with r, s being the order of spatial and temporal accuracy, respectively.

A second requirement concerns to the stability of the scheme and estab-lishes a relation between the computed and the exact solutions of the discre-tised equation. As introduced by Lax and Richtmyer [16], the stability criterionstates that any component of the initial solution should not be amplified with-out bound. Following the procedure given by Hirsch [13], consider a marchingproblem in which at a certain time n the variable un is known. The discretescheme can be represented as

un+1 = Cun or un+1 = (C)nu0 (2.22)

where C is a discrete operator which when applied to un returns a value forthe unknown un+1 at time level n+1, (C)n represents the operator C applied ntimes, assuming that the superposition principle is valid. The stability condi-tion for the scheme represented by the operator C can be achieved if a constantK exists, such that

‖ (C)n ‖< K for0 < ∆t < τ

0 ≤ n∆t ≤ T(2.23)

for fixed values of τ , T and for all n, with ‖ • ‖ denoting an appropriate norm.

The analysis of stability can be accomplished by various methods, such asthe Von Neumann method, the equivalent differential equation and the matrixmethod [13]. Each method has its own merits but all result from linear theory


PartialDifferentialEquations

DiscretizedEquations

Exact Solution ofPartial DifferentialEquations

ClosedForm

Discretization

Consistency

Sta

bilit

y

Com

puta

tion

ConvergenceNumerical Solution ofDiscretized Equations

Exact Solutionof DiscretizedEquations

ClosedForm

Path 1

Path 2

Figure 2.9: Relations between consistency, stability and convergence.

and only represent a rational support and guideline for non–linear problems,for which the last word will be given to numerical experiments.

The primary requirement of a numerical scheme is that the numericalsolution must approach the exact solution of the differential equation at anypoint and at any time when ∆x, ∆t tend to zero. This condition is calledconvergence and, despite being very difficult to establish directly, it is auto-matically achieved once consistency and stability are verified as result of thefundamental Lax’s equivalence theorem [26], which may be stated as “For awell–posed linear initial value problem and a consistent discretisation, stabilityis necessary and sufficient condition for convergence”.

To summarize these concepts, see figure 2.9 where a sketch showing theconceptual relations between consistency, stability and convergence is pre-sented and where the indirect route to achieve convergence is represented bypath 2.


2.4 Conservation Laws & Constitutive Equa-

tions

The majority of the phenomena encountered in fluid mechanics falls well withinthe realm of the continuum postulate and the physical description of our worldcan neglect the phenomena occurring in a microscopic level. Although thiswould appear to be less realistic than the kinetic theory approach, it is farsimpler to use the phenomenological approach, with the continuum model, toderive the equations of fluid dynamics.

The continuum assumption requires that the mean free path of individualelements must be very small when compared with the smallest physical–lengthscale of the system under consideration, i.e. the density of elements is highenough such that the mutual interaction dominates over the individual be-havior which is in general meaningless. Where the microscopic–length scaleapproaches macroscopic dimension, such as when rocket passes through theedge of the atmosphere, in which rarefied gas exist, the interaction betweenparticles becomes insignificant and the particles behave essentially as individ-ual elements. These limit situations are outside the field of fluid dynamics.Where the smallest macroscopic–length scale approaches microscopic dimen-sions, such as in the structure of a shock wave or phase interfaces, once morethe continuum hypothesis is no longer valid but a weak solution is still possibleas will be discussed later when considering hyperbolic equations.

The basic conservation laws in fluid mechanics can be derived in the Eu-lerian framework by considering the fluid which pass at a time t through anarbitrary fixed control volume V with surface S in relation to a fixed cartesiansystem of reference as described in figure 2.10. The equations presented inthis section are derived assuming a single–phase, homogeneous fluid in whichno chemical reaction is taking place. An indicial notation is adopted, withthe summation convention implying summation over any repeated index in aterm of an expression [29]. The reader is referred to the available literature[1, 5, 11, 13, 28, 30], for a more detailed discussion on this theme.

2.4.1 Conservation of mass

The fluid mass in the control volume V can change only because of massflowing across the surface S . Mathematically, this statement may be writtenas

∫

V

∂ρ

∂tdV = −

∫

SρujnjdS (2.24)


n

V

Sds

x1

x3

x2

Figure 2.10: Eulerian coordinates and control volume.

where ρ represents the fluid density, t is the independent time variable, uj isthe fluid velocity component and nj is the component of unit outward nor-mal to S , with the index j denoting the spatial dimension xj . Assuming anappropriate level of smoothness for the integrand in the right hand side ofequation (2.24), the divergence theorem can be used and the mass conserva-tion statement becomes

∫

V

[∂ρ

∂t+

∂(ρuj)

∂xj

]dV = 0 (2.25)

or, as the control volume V was arbitrarily chosen,

∂ρ

∂t+

∂(ρuj)

∂xj

= 0 (2.26)

everywhere in the domain V . This statement is usually called the continuityequation.

2.4.2 Conservation of momentum

Excluding mutual forces, such as electric forces due to electrostatic chargesinside the fluid, the application of Newton’s second law of motion to a fluidpassing through a fixed control volume yields, in the direction xi, the momen-tum conservation equation


∫

V

∂(ρui)

∂tdV = −

∫

SρuiujnjdS +

∫

SnjσijdS +

∫

VρfidV (2.27)

where the rate of change of fluid momentum in V is equal to the rate ofmomentum flow across S plus the net external forces exerted on the massof fluid in V . These forces are represented either as body forces fi, such asgravitational or electromagnetic forces, or surface forces which are representedby the components of the stress tensor σij . For the class of fluids consideredhere, see discussion in section 2.4.4, the stress tensor has the form

σij = −pδij + τij (2.28)

where δij is the Kronecker delta, p is the thermodynamic pressure and τij isthe shear–stress tensor. Making use of this stress relation, assuming that thequantities involved in the surface integrals in equation (2.27) are differentiableand using the divergence theorem, the equation of momentum conservationcan be rewritten as

∫

V

[∂(ρui)

∂t+

∂(ρuiuj)

∂xj+

∂(pδij)

∂xj− ∂τij

∂xj− ρfi

]dV = 0 (2.29)

From the arbitrariness of the choice of the control volume, the momentumequation becomes

∂(ρui)

∂t+

∂

∂xj

(ρuiuj + pδij

)=

∂τij

∂xj+ ρfi (2.30)

everywhere in the domain V .

2.4.3 Conservation of energy

The specific energy ε of a fluid consists of the specific internal energy e plusthe kinetic energy, and may be expressed as

ε =1

2uiui + e (2.31)

With this in mind, ignoring any effect of energy transmission by radi-ation from an external body or within a hot gas stream, the mathematicalrepresentation of the first law of thermodynamics can be stated as


∫

V

∂(ρε)

∂tdV = −

∫

SρεujnjdS +

∫

SnjσijuidS +

∫

VρujfjdV

−∫

SqjnjdS +

∫

VQdV

(2.32)

where the rate of change of the total energy of the fluid as it flows is equal tothe sum of the total energy that flows across S , the total work done on thefluid by external forces, the net amount of heat transfer by conduction andby a source or sink of heating. The conduction heat flux in direction xj isdenoted by qj and the source or sink of heating per unit of volume is givenby Q. By applying the divergence theorem to the surface integrals, equation(2.32) becomes

∫

V

[∂(ρε)

∂t+

∂(ρεuj)

∂xj− ∂(σijui)

∂xj+

∂qj

∂xj− ρujfj − Q

]dV = 0 (2.33)

The integral must be zero for any arbitrary control volume V , and so theintegrand is zero and the conservation of energy equation, making use of (2.28)and after rearranging, takes the form

∂(ρε)

∂t+

∂

∂xj

[(ρε + p)uj

]=

∂(τijui)

∂xj

− ∂qj

∂xj

+ ρujfj + Q (2.34)

everywhere in the domain V .

2.4.4 Constitutive equations

The conservation laws, described previously, represent a mathematical for-mulation of the physical principles of conservation and therefore apply to allmaterials. To complete the specification of the mechanical properties of amaterial or classes of materials some additional equations, which are calledconstitutive equations, are required. Although, it is unlikely that any real ma-terial will conform exactly to any mathematical model, we can produce modelswhich form an excellent approximation to the behavior of the real material.The constitutive equations must satisfy fundamental principles, such as theprinciple of material frame-independence, the principle of determinism, theprinciple of local action and also be dimensionally consistent ([11, 28, 33]).Further axioms must be stated to fully define a constitutive equation for anspecific material or class of materials.


In fluid mechanics, axioms which allow us to determine the relations be-tween stress and rate of strain, and heat flux and temperature are required.The first constitutive relation relating the stress tensor σij and the rate ofstrain tensor ǫij can be deduced with the assumption of the following fourpostulates ([28, 30]):

1) The stress tensor is linearly dependent on the deformation-rate tensor ǫkl.This behavior is characteristic of a Newtonian viscous fluid and is mathemat-ically expressed by

σij = aij + bijklǫkl (2.35)

where aaa, bbb represent a second and fourth–order tensors and are independentof the rate of strain components;

2) In a fluid at rest, the stress is hydrostatic and the pressure exerted by thefluid is the thermodynamic pressure p. This condition requires that the stresstensor defined in equation (2.35) must have a form

σij = −pδij + τij (2.36)

where the shear-stress tensor τij depends upon the motion of the fluid;

3) The shear stresses τij will not be affected by rigid-body motion of the fluid;

4) The fluid is isotropic, i.e. there are no preferred directions in the fluid.

After all possible simplifications using these assumptions, the constitutiveequations for the stress tensor reduces to

σij = −pδij + λδijǫkk + µǫij (2.37)

where p, λ and µ are independent of ǫij . The rate of strain tensor is describedin terms of velocity variations according to

ǫij =∂ui

∂xj+

∂uj

∂xi(2.38)

The parameters µ, λ must be determined experimentally and representthe dynamic viscosity and second viscosity coefficients, respectively. Thesetwo viscosity coefficients are related through the bulk viscosity χ ([1, 30]), andfor monotonic gases, χ is negligible (χ ≈ 0), leading to the requirement that

λ = −2

3µ (2.39)


which represents the so called Stokes’ relation. In practice, it is found that themodel given by equations (2.37) to (2.39), despite the fact that it can be re-garded as describing an ideal material, simulates extremely well the mechanicalbehavior of many fluids, including air and water in most common situations.

The second constitutive relation involves a relation between the conduc-tive heat–flux qj and the temperature gradients. A simple, but accurate, con-stitutive equation is given by Fourier’s law for heat transfer by conduction,which can be expressed as

qj = −k∂T

∂xj(2.40)

where k = k(T ) is referred to as the thermal conductivity of the medium, andT is the temperature.

2.5 Fluid Dynamics and Heat Transfer Equa-

tions

The set of equations involving mass (2.26), momentum (2.30) and energy (2.34)conservation in a continuous medium is commonly referred to as the Navier-Stokes equations. These equations, in the conservation form, can be writtenusing a compact vector notation as

∂U

∂t+

∂Fj

∂xj=

∂Gj

∂xj+ Bj + S for j = 1, . . . Nd (2.41)

where the Nd indicates the number of spatial dimensions to be considered inthe model. In equation (2.41), U is the vector of the conservative variables,

while the vectors Fj, Gj and Bj denote the inviscid flux, the viscous flux and

the body forces actions in the direction xj , respectively, and S is the vectorwhich takes into account the source of heating term. Considering the twodimensional counterpart of equation (2.41), these vectors can be written as

U =

ρρu1

ρu2

ρε

F

j=

ρuj

ρu1uj + pδ1j

ρu2uj + pδ2j

(ρε + p)uj

Gj =

0τ1j

τ2j

uiτij − qj

(2.42)

Bj =

0ρf1

ρf2

ujfj

S =

000Q

(2.43)


where τij and qj are defined by the two constitutive equations presented insection 2.4.4, and re–written as

τij = µ

(∂ui

∂xj+

∂uj

∂xi

)− 2

3µ

∂uk

∂xkδij and qj = −k

∂T

∂xj(2.44)

These equations are valid for any Newtonian compressible fluid and will alsobe used to deduce the particular equations for different dynamic levels of ap-proximation analysed in the present work, as well as, heat conduction in astationary medium.

2.5.1 The gas dynamics equations

When the application, of the equations of motion involves flow speeds compa-rable with that of the speed of sound, which is generally accompanied by largepressure gradients leading to substantial changes in density, compressibilityeffects must be considered [2]. These effects are of practical importance forgases, such as in aerodynamics applications. Let us consider the 2–D modelof a compressible flow without body forces (B ≡ 0) and without external orinternal heat sources (S ≡ 0), described by equations (2.41 – 2.44). In order toclose the system of equations it is necessary to set up the relations between thethermodynamics variables through the state equations and to relate the coef-ficient of viscosity µ and the thermal conductivity k to the thermodynamicsvariables.

It is convenient to introduce the thermodynamic variable called enthalpyh, which is defined as

h = e +p

ρ(2.45)

Consider an ideal gas, where no real gas effects or chemical reactions oc-cur, the fundamental gas relationship, derived from kinetic theory [18], relatespressure with density and temperature according to

p = ρRT (2.46)

where R is called the “gas” constant. Assuming further that the gas is calor-ically perfect, where the internal energy e and enthalpy h are functions oftemperature alone and the specific heats are constants, then

e = CvT h = CpT γ = Cp/Cv Cv = R/(γ − 1) (2.47)


where Cp and Cv are the specific heats of the fluid at constant pressure and atconstant volume respectively. Combining equation (2.31) and equations (2.45)to (2.47), the state equations can be written as follows

p = (γ − 1) ρ(ε − 1

2uiui

)T =

p

cv(γ − 1)ρ(2.48)

Sutherland’s experimental law relates the dynamic coefficient of viscosityµ to the temperature according to

µ = µr(Tr + S0)

(T + S0)

[T

Tr

]3/2

(2.49)

where the subscript r denotes a reference state, which is usually given by thefree stream values and S0 is a constant for a given gas [13]. The coefficient ofthermal conductivity k can be related to the coefficient of viscosity throughthe definition of the dimensionless quantity

Pr =cpµ

k(2.50)

known as Prandlt number, which is assumed constant for ideal gases at mod-erate temperature [18].

For an isentropic process, i.e. adiabatic reversible process [2, 18], in aperfect gas, the speed of sound c is given by

c2 =

(∂p

∂ρ

)

s

=γp

ρ(2.51)

where s is the entropy and c represents the speed at which the small distur-bances (waves) are propagated through a compressible fluid. The effects ofcompressibility in a flowing fluid can then be analysed, by the value of thedimensionless parameter

M =(ujuj)

1/2

c(2.52)

which is called Mach number, and its variation significantly alters the characterof a flow. Finally, another important dimensionless relation is given by theReynolds number

Re =ρ|u|L

µ(2.53)


where L is a characteristic length of the problem, such as an aerofoil chordlength, and |u| is the velocity modulus. The Reynolds number representsthe ratio of the inertial terms and the viscous terms, with the flow remaininglaminar up to a certain critical value of the Reynolds number and above thiscritical value the flow becomes turbulent.

In some applications associated to hypersonic flows, where temperaturescan be extremely high, the perfect gas assumption of a gas which is not chem-ically reacting and calorically perfect, does not apply anymore. Furthermore,the assumption of constant Prandlt number and Sutherland’s law do not holdeither. In this regime, more complex state equations and relations with thetransport coefficients have to be adopted [19, 37].

Non–dimensional form of equations

The governing fluid dynamic equations are often rewritten in a non-dimensionalform. This form has practical implications as it permits the full characteriza-tion of a flow by the definition of some dimensionless parameters. A varietyof different non–dimensional forms of the variables can be adopted [1, 6, 34].Denoting a non–dimensional variable by the use of an asterisk as a superscriptand the freestream values by the subscript ∞, the following relations [34] areadopted in the present work

x∗ = xj/L u∗j = uj/u∞ ρ∗ = ρ/ρ∞

µ∗ = µ/µ∞ k∗ = k/k∞ t∗ = u∞t/L

p∗ = p/ρ∞u2∞ ε∗ = ε/u2

∞ T ∗ = cpT/u2∞

(2.54)

Here, L is a representative length scale. The speed of sound and Mach number,given in equations (2.51) and (2.52) respectively, have their freestream valuescomputed according to

c2∞ = γp∞/ρ∞ M∞ = u∞/c∞ (2.55)

which allow us to determine the non–dimensional freestream pressure and tem-perature as

p∗∞ = 1/γM2∞ T ∗

∞ = 1/(γ − 1)M2∞ (2.56)

The freestream Reynolds number is defined as

Re∞ =u∞ρ∞L

µ∞

(2.57)


After applying this non–dimensionalizing procedure to the Navier–Stokesequations, given by equations (2.41) to (2.44), with no body force or sourceterm, the system of non–dimensional equations may be conveniently writtenas

∂U∗

∂t∗+

∂Fj∗

∂x∗j

=∂Gj∗

∂x∗j

for j = 1, . . . Nd (2.58)

where the vector of conserved variables U∗, the convective flux vectors Fj∗

and the viscous flux vectors Gj∗, in two dimensions, are given by

U∗ =

ρ∗

ρ∗u∗1

ρ∗u∗2

ρ∗ε∗

F

j∗=

ρ∗u∗j

ρ∗u∗1u

∗j + p∗δ1j

ρ∗u∗2u

∗j + p∗δ2j

(ρ∗ε∗ + p∗)u∗j

Gj∗ =

1

Re∞

0τ ∗1j

τ ∗2j

u∗i τ

∗ij + µ∗

Pr

∂T ∗

∂x∗

j

(2.59)

In this expression the heat–flux vector in non–dimensional form has alreadybeen incorporated in the vector Gj∗. The shear–stress tensor components aredefined by

τ ∗ij = µ∗

(∂u∗

i

∂x∗j

+∂u∗

j

∂x∗i

)− 2

3µ∗∂u∗

k

∂x∗k

δij (2.60)

The non–dimensional perfect gas equations of state (2.48) becomes

p∗ = (γ − 1) ρ∗

(ε∗ − 1

2u∗

i u∗i

)T ∗ =

γp∗

(γ − 1)ρ∗(2.61)

Finally, introducing the non–dimensional variables in the Sutherland re-lation (2.49) gives

µ∗ =

[(γ − 1)M2

∞]−1

+ S∗

T ∗ + S∗

[(γ − 1)M2

∞T ∗]3/2

(2.62)

with

S∗ =S0

M2∞T∞

(2.63)

In the remainder of this work, the non–dimensional form of the equationswill be employed. However, for notational convenience, the asterisk nota-tion will be dropped on the understanding that all variables are now non–dimensional. It can be observed that, for any set of boundary conditions, the


flow is characterized by the dimensionless parameters Re∞, Pr, M∞, the refer-ence temperature T∞ and the angle between the freestream flow and the maindirection of the body, e.g. the chord line of an aerofoil. This angle is normallyreferred to as the angle of attack or the angle of incidence.

The compressible Euler equations

Although no fluid in nature is actually inviscid, when viscid–inviscid interac-tions are negligible, the inviscid assumption leads to a good approximationof the pressure field and hence of the lift coefficient for non-separated flows[2, 14]. This represents a large range of aerodynamic flows. This approxima-tion has its basis in Prandtl’s boundary layer analysis, which shows that for aflow with no separation and at high Reynolds number, the viscous and turbu-lent effects have their influence confined to narrow regions close to the walls.Outside these layers, the flow behaves as inviscid. The inviscid considerationis motivated by the simplified form of the resulting governing equations and bythe advantages with regard to the reduced computational requirements whencompared with those of the viscous model.

The set of Euler equations in non–dimensional conservation form is ob-tained from the Navier–Stokes equations given in (2.58) by neglecting all shear–stress and heat conduction terms (Gj ≡ 0),

∂U

∂t+

∂Fj

∂xj= 0 for j = 1, . . .Nd (2.64)

The system is closed by the addition of the state equation

p = (γ − 1) ρ(ε − 1

2uiui

)(2.65)

which relates the pressure, density and energy in the flow of an ideal gas.

A major change in the mathematical model character results from this ap-proximation, since the system of equations describing the inviscid flow reducesfrom second–order to first–order. Such system is totally hyperbolic. This notonly influences the physical behavior and the definition of boundary condi-tions for the model, but also the design of numerical schemes to solve the flowsdescribed by this model.

2.5.2 The incompressible fluid flow equations

If a given fluid is comparatively difficult to compress, requiring a very largepressure increase to produce a small decrease in volume, it may be reasonable


to assume total incompressibility of the medium. This condition is normallyadopted for the simulation of flow of liquids. Also, if gas velocities are suffi-ciently low when compared with the local speed of propagation of the sound,and the temperature variations are small when compared with the average ab-solute temperature of the gas, then it may be satisfactory to assume the gasto be incompressible [22].

The incompressible assumption requires the density ρ to be constant andthe first row of equation (2.59), which represents the continuity equation, canbe simplified to

∂uj

∂xj= 0 (2.66)

Making use of this equation, the components of the shear–stress tensor givenby equation (2.44) reduce to

τij = µ

(∂ui

∂xj+

∂uj

∂xi

)(2.67)

and the Stokes’ relation is immaterial as the term involving λ vanishes.

In most frequently encountered situations, the dynamic viscosity may beconsidered constant, and this together with the continuity equation allowsfurther simplifications on the conservation of momentum equations given bythe second and third rows of equation (2.59). For the 2–D model consideredhere, they can be rewritten as

∂(ρui)

∂t+

∂(ρuiuj)

∂xj+

∂(pδij)

∂xj=

µ

Re∞

∂2ui

∂xj∂xj+ ρfi (2.68)

where the body forces terms, (2.41), was added.

Once the density and viscosity of a fluid are specified, the set of equationsgiven by (2.66) and (2.68) is enough to determine the pressure and velocityfields without reference to the energy equation. If required the energy equa-tion can be utilized later to determine the temperature distribution. Thismathematical un-coupling of the continuity and momentum equations fromthe energy equation is adopted, as thermal effects are unimportant for theincompressible flows considered here. Furthermore, since very low fluid veloc-ities with continuous solution are analysed with incompressible simulations,the equations of motion can be written in non–conservative form as a functionof the primitive variables ρ, uj and p according to

∂uj

∂xj= 0 (2.69)


ρ∂ui

∂t+ ρuj

∂ui

∂xj+

∂p

∂xi=

µ

Re∞

∂2ui

∂xj∂xj+ ρfi (2.70)

These equations describe the motion of an incompressible Newtonian fluid withconstant viscosity. The incompressibility assumption has many implications,not only in the physical concepts involved and the mathematical model whichdescribes the flow behavior, but also in the nature of the numerical algorithmsdeveloped for their simulation.

2.5.3 The heat transfer equation

One of the simplest class of problems involving the conservation laws is thestudy of heat conduction in a stationary medium (uj = 0). Here, no attentionis paid to the stress distribution in a heated medium and such problems canbe modeled using only the balance of energy equation, which is described inthe fourth row of equation(2.59). In this case, this equation can be rewrittenas

∂(ρε)

∂t+

∂qj

∂xj

= Q (2.71)

In a stationary incompressible fluid the relations

ε = e ρ = const. p = const. (2.72)

are valid and, if the fluid is calorically perfect, the specific energy e and theenthalpy h are functions only of the temperature, as discussed in section 2.5.1.By using the enthalpy definition given in (2.45), together with relations pre-sented in (2.72), we get

Cv =∂e

∂T=

∂ε

∂T=

∂h

∂T= Cp (2.73)

This means that there is no distinction between heat capacities, and so

∂(ρε)

∂t= ρ

∂e

∂T

∂T

∂t= ρCp

∂T

∂t(2.74)

Making use of the previous relations and of the Fourier’s law for heat conduc-tion (2.40), the energy equation (2.71) can be rewritten as

ρCp∂T

∂t− ∂

∂xj

(k

∂T

∂xj

)= Q (2.75)


This is the parabolic partial differential equation which governs the conductionof heat in a stationary fluid, and also in a solid, since all considerations usedin the derivation of equation (2.75), i.e. incompressible, stationary, caloricperfect medium, can also apply to a solid.

2.5.4 Initial and boundary conditions

It is necessary to set suitable initial and boundary conditions for a completedefinition of the mathematical model which governs the flow of a viscous com-pressible fluid and the various simplified models derived in previous sections.

The initial condition is specified by assuming that

U(xj , t0) = U0(xj) for all xj in Ω at time t = t0 (2.76)

Here U0 represents a known function and is normally taken as the undisturbedflow (freestream regime, U∞) in aerodynamics applications.

The concepts introduced in section 2.2 to classify the mathematical na-ture of a partial differential equation, also apply individually to each equationthat belongs to a system of partial differential equations. Therefore, a cou-pled system of partial differential equations can exhibit simultaneously elliptic,parabolic and hyperbolic characteristics. The Navier–Stokes set of equationsrepresents a hybrid system, being parabolic–hyperbolic in time and space, butbecoming of mixed elliptic–hyperbolic nature in space for the stationary for-mulation. No general mathematical theorems concerning the proper boundaryconditions, that ensures existence and uniqueness of the solution, are available[7, 12]. In this way, the approach normally adopted consists of the physicalanalysis of the problems to provide a guide line for the choice of number andtype of boundary conditions, followed by an a posteriori numerical validation.

In practical applications, the solution of partial differential equations isdevised in a limited domain and, apart from physical boundary conditionssome artificial boundary conditions must be imposed to produce a boundeddomain, which are normally referred to as numerical boundary conditions. Ingeneral, the boundaries can be physically grouped into two categories, eithersolid body surfaces, such as an airplane immersed in a flow, or as free surfaces,such as far–field boundaries in external flows. In figure 2.11 a typical finitedomain for the analysis of a simulated forebody is presented where the twodistinct groups are clearly recognized. In this figure the vectors n = (n1, n2)and t = (t1, t2) represent the normal and tangential directions on the boundaryrespectively. The domain is represented by Ω, the solid wall surface by Γw andthe free surface by Γf .


x2

x1

Γf

Free StreamFlow

n t wΓ

tn

Ω

Figure 2.11: Finite domain and boundary configuration for an external flow.

When defining the free surface boundary conditions, it is also useful to dis-tinguish internal and external flows, and to sub–divide the free surface bound-ary into two distinct parts, viz the inflow boundaries, through which the flowenters the domain, and the outflow boundaries, through which the flow leavesthe domain. This partition is determined according to the sign of the nor-mal velocity across the boundaries. Moreover, the boundary conditions aredefined depending on the local flow regime, which may be either subsonic orsupersonic.

Different levels of complexity can be incorporated in the determination ofproper free surface boundary conditions for the Navier–Stokes equations. Apossible approach for external flow computation, provided that the far fieldboundary is placed far from the body, so that viscous effects can be assumednegligible is to determine the boundary conditions through the analysis ofthe incoming and outcoming characteristics [14] of the Euler equations set.Another simple, but normally acceptable procedure, especially for supersoniccomputations, prescribes all flow variables through the inflow boundaries andfor the outflow, considering the fact that the flow is strongly dependent onits evolution inside the computational domain, just leaves all flow variablesfree to change. When the free surface occurs close to the solid wall or, asin the case of an internal flow, the viscous effects can not be ignored, theappropriate outflow boundary condition is not yet theoretically established.


More appropriate conditions can be devised and a detailed discussion aboutthis essential subject can be found in [7, 14, 23, 27, 32].

At solid body surfaces or simple solid walls no relative velocity betweenthe fluid and a solid boundary at the interface is observed experimentally. Ona microscopic level, a slippage is possible [5], but this is out of the domain ofcontinuum mechanics. Then, the no–slip condition is assumed to hold and canbe stated as

u = 0 at Γw (2.77)

where u represents the velocity vector with components ui.

The other boundary conditions at a wall are normally of either Dirichletor Neumann type, and include the case where either the temperature of theflow at the wall Tw is known,

T = Tw at Γw (2.78)

or the wall heat flux qw is prescribed as

k∂T

∂n= qw at Γw (2.79)

The most common example of the first type is an isothermal wall, where Tw isconstant, and for the second type is an adiabatic wall, where the heat flux qw

is equal to zero.

It should be mentioned that the simplified cases represented by the Eu-ler equations or the heat conduction equation are mathematically fully de-termined, and that the adopted boundary conditions for viscous simulationshave to be compatible with the conditions used for inviscid computations inthe limit of vanishing viscosities, otherwise, non–physical behaviour may ap-pear in the solution [14, 23]. Furthermore, since the problems considered inthis work have a distinct mathematical nature and physical properties, theadopted boundary conditions for each specific case will be presented in detailwhen dealing with each problem. Finally, the complexity and importance ofthe theme cannot be underestimated, and one has to be aware of the influenceof the selected conditions on the final stability, convergence rate and accuracyof the numerical solution.

2.6 General Remarks

Some generic issues concerning the difficulties normally faced, when design-ing a computational procedure in computational fluid dynamics, should bementioned.


2.6.1 Mathematical difficulties

It is far from an easy task establishes the well–posedness of IBVPs, especiallyfor systems of partial differential equations of mixed nature, where no gen-eral mathematical theorem to guarantee existence and uniqueness of solutionis known. In general, an under-prescription of boundary conditions leads tonon–uniqueness and an over-prescription leads to unphysical solutions in thevicinity of the boundary in question [10]. There are some flow problems forwhich multiple physically meaningful solutions may be expected and the con-cept of well–posedness fails. This situation often occurs for flows undergoingtransition from laminar to turbulent motion[10]. Also, the lack of mathemat-ical foundations establishing consistency, stability and convergence for non–linear problems leaves only the well–based linear theory to give necessary, butnot sufficient, conditions on the behavior of the numerical scheme.

Another mathematical difficulty arises from the possibility of the appear-ance of discontinuous solution when hyperbolic equations are considered. It isobvious that such a solution does not satisfy the PDE in the classical sense,as derivatives are not defined at discontinuities. But, as was shown in section2.4, the physical derivation of the conservation laws leads to an integral equa-tion and the differential equation can only be obtained by imposing additionalsmoothness assumptions. The integral form is still valid, even for discontin-uous solutions. Another approach to deal with discontinuous solutions is theuse of the weak form of the differential equation as introduced in section 2.3.3,which also involves integrals. The weak form is fundamental in the develop-ment and analysis of numerical methods. Nevertheless, the use of a weak formoften allows more than one solution to a problem with the same boundaryand initial conditions. The fact that non–linear hyperbolic equations, suchas the inviscid Burgers equation or Euler system of equations, admit spurioussolutions is in general a result of ignoring some physical effects in the determi-nation of the model, i.e. the diffusive or viscous effects have been neglected.Although diffusive effects may be negligible throughout most of the flow, neardiscontinuities the effect is always strong, and the apparent discontinuities arein reality thin regions with very steep gradients. Some conditions must beimposed in order to pick up the correct physical solution and to guaranteeuniqueness. In fluid dynamics the second law of thermodynamics, which statethat entropy should increase, is invoked and turns out to be a sufficient con-dition to determine the physically correct and specify a unique solution. Thisconsideration is normally referred to as an entropy condition [13, 17].

2.6.2 Numerical difficulties

Once the mathematical model is established, taking into account appropri-ate levels of approximation to the physical phenomena (see sections 2.4 and


2.5), which is dictated by the accuracy required and the computational poweravailable, the followings general steps are necessary in the development of acomputational code to numerically simulate the adopted model.

The first step is achieved by replacing the infinite number of points exist-ing on the continuum space, by a finite number of points, which will representthe phenomena in the discrete domain. This stage is accomplished by the use oftechniques to generate meshes or grids on a given finite domain, and the basicrequirements are the correct representation of the geometry boundary of thedomain and the flexibility for the proper distribution of the points according tothe main physical features of the problem analysed [35]. The complex geome-tries and flow features involved in computational fluid dynamics applicationsmake this stage an essential pre–requisite for a successful simulation.

Then, the system of partial differential equations can be discretised inspace. This is accomplished by the use of one of the available techniques,such as the Finite Difference Method, the Finite Volume Method, the FiniteElement Method, etc [13]. The importance of continuous development in thisarea can not be overemphasized since it represents the core of numerical sim-ulation. Certain difficulties which are faced at this stage can be mentioned.It should be observed that the presence of the convective term in fluid dy-namics applications breaks the symmetry of the differential operator, which isnon self–adjoint [24], and the utilization of naive methods developed for ellip-tic problems gives disastrous results, mainly in the presence of discontinuitiesor when incompressibility assumption is adopted. Also, for non–linear prob-lems, with discontinuous solutions, the numerical method might converge to afunction that is not a weak solution of the original equation.

For time–dependent problems, or when a pseudo–transient approach isadopted for solving stationary problems, a system of ordinary differential equa-tions will result from the previous step. This system should be further discre-tised in time leading to an algebraic, linear or non–linear, system of equationsat a given time level. Two families of techniques, explicit or implicit, are avail-able to accomplish the time discretisation and a large number of methods tosolve the system of algebraic equations or to accelerate these procedures can beemployed. The choice of methodologies in this stage will in general determineboth the memory requirements and the CPU time involved in the simulation,so representing a paramount stage concerning the efficiency of the code.

As a final step, the code must be provided with some techniques thatpermit an assessment and control on the quality of the resulting approximatesolution. The estimation of the error due to the discretisation and the self–adaptive correction are of utmost importance, providing a rational way toaddress challenging applications.

Bibliography


[2] Jr. ANDERSON, J.D. Modern Compressible Flow. MacGraw–Hill, 1990.

[3] K.J. BATHE. Finite Element Procedure in Engineering Analysis.Prentice–Hall, Inc., 1982.

[4] E.R. BENTON and G.W. PLATSMAN. A Table of Solutions of the One–Dimensional Burgers Equation. Quart. Appl. Math., 30:195–212, 1972.

[5] I.G. CURRIE. Fundamental Mechanics of Fluids. MacGraw–Hill, 1974.

[6] K.N. DANTJE. Finite Volume Solutions of the Two–Dimensional Com-pressible Flow Equations on Structured, Unstructured and Hybrid Grids.PhD thesis, University College of Swansea, 1992.

[7] P. DUTT. Stable Boundary Conditions and Difference Schemes forNavier–Stokes Equations. SIAM J. Numer. Anal., 25(2):245–267, 1988.

[8] L.E. ELSGOLC. Calculus of Variations. Pergamon Press, 1961.

[9] S.J. FARLOW. Partial Differential Equations for Scientists & Engineers.John Wiley & Sons, 1982.

[10] C.A.J. FLETCHER. Computational Techniques for Fluid Dynamics, vol-ume 2. Springer–Verlag, 1988.

[11] M.E. GURTIN. An Introduction to Continuum Mechanics. AcademicPress Limited, 1981.

[12] B. GUSTAFSSON and A. SUNDSTROM. Incompletely Parabolic Prob-lems in Fluid Dynamics. SIAM J. Appl. Math., 35(2):343–357, 1978.


48




[16] P.D. LAX and R.D. RICHTMYER. Survey of the Stability of Linear Fi-nite Difference Equations. Comm. Pure and Applied Mathematics, 17:267–293, 1956.

[17] R.J. LE VEQUE. Numerical Methods for Conservation Laws. BirkhauserVerlag, 1990.

[18] H.W. LIEPMANN and A. ROSHKO. Elements of Gasdynamics. JohnWiley & Sons, Inc, 1957.

[19] M.-S. LIOU, B. VAN LEER, and J.S. SHUEN. Splitting of Inviscid Fluxesfor Real Gases. J. Comput. Physics, 87:1–24, 1990.

[20] K. MORGAN, J. PERAIRE, and J. PEIRO. Unstructured Grid Methodsfor Compressible Flows. In AGARD Report 787 on Special Course onUnstructured Grid Methods for Advection Dominated Flows, pages 5.1–5.39, 1992.

[21] J.T. ODEN. Applied Functional Analysis: An Introductory Treatementfor Students of Mechanics and Engineering Scence. Prentice–Hall, 1979.

[22] J.E. PLAPP. Engineering Fluid Mechanics. Prentice–Hall, 1968.

[23] T.J. POINSOT and S.K. LELE. Boundary Conditions for Direct Sim-ulations of Compressible Viscous Flows. J. Comp. Phys., 101:104–129,1992.

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[25] L.F. RICHARDSON. Mathl. Gaz., 425, 1925.

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[28] J.C. SLATTERY. Momentum, Energy and Mass Transfer in Continua.Robert E. Krieger Publishing Company, Huntington, New York, 1981.


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[32] K. W. THOMPSON. Time Dependent Boundary Conditions for Hyper-bolic Systems. J. Comp. Phys., 68:1–24, 1987.

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Chapter 3

Adaptive Procedure forTransient Heat ConductionSimulation

3.1 Introduction

The reliability of an approximate solution computed using a numerical tech-nique is directly related to the capacity of the discrete model to represent thephysical behavior described through the mathematical model under considera-tion. Assuming that an appropriate numerical technique is in hand, two otherrequirements must be fulfilled to obtain a reliable result. The first requirement,is to adequately represent the geometry of the computational domain with adiscrete grid. This task requires the development of suitable mesh generatorsand is clearly independent of the problem solution. Since it is unacceptablyinefficient to construct a fine discretisation everywhere, it is necessary to con-nect the discretisation with the solution. This represents the goal of the secondrequirement, which refers to the design of some techniques to permit an as-sessment of the error in the solution combined with a full adaptive process toimprove continuously the discrete model.

There is a general agreement that adaptivity will become a standard fea-ture of computer software in the near future. Substantial progress has beenmade toward this direction, however, there are numerous problems to be over-come if adaptivity is to be used with confidence in the solution of practi-cal problems. The development of adaptive procedures for elliptic problems,where solely the space is adapted, has received a lot of attention and hasachieved a quite mature stage. The smoothness of the solution of parabolicproblems varies considerably in space and time, exhibiting for instance, initialtransients where highly oscillatory components of the solution are rapidly de-caying. Therefore, efficient computational methods, for this class of problems,

Heat Conduction 53

requires the use of mesh spacing and time–steps which are variable, ideally inboth space and time.

In this chapter, one possible finite element adaptive procedure based uponthe use of mesh enrichment and time–step control, dictated by the use of aposteriori error estimators, is presented for the solution of the scalar parabolicequation which governs heat conduction in a stationary medium. Some prac-tical implications of simultaneously adapting the mesh spacing and time–stepsize are discussed.

3.2 Numerical Solution Procedure

Considering the Fourier law of conduction in a continuous medium, the 2–D transient heat conduction problem, in the absence of internal or externalradiation, is described in section 2.5.3 and rewritten here as

ρCp∂T

∂t− ∂

∂xj

(k

∂T

∂xj

)= Q in Ω × I (3.1)

subject to the boundary conditions

T = T (t) at ΓD × I

k∂T

∂n= q(t) at ΓN × I

(3.2)

and the initial condition

T = T0

on Ω × t0 (3.3)

where the boundaries ΓD and ΓN are assumed not to vary in time, T and q arethe prescribed, exact, temperature and heat flux. The medium is consideredisotropic with ρ, Cp, k constants and (3.1) represents a linear non–homogeneousparabolic second–order partial differential equation. The inclusion of domainand boundary radiation terms represents no additional problem to the wholeprocedure described in this chapter [28].

3.2.1 Spatial discretisation

Assume that the spatial domain Ω is discretised into an assembly of bi–linear4–noded isoparametric finite elements ΩE , with nodes numbered from 1 top. The subsets TTT (p) and WWW(p) of the trial and weighting function sets, (see

Heat Conduction 54

section 2.3.2), for use with a Galerkin weighted residual approximate method,are defined by

TTT (p) = T (xj , t) | T =p∑

J=1

TJ(t)NJ (xj); TJ(t0) = T0(xj) = T 0

J

WWW (p) = W(xj) |W =p∑

J=1

aJNJ(xj)(3.4)

where NJ is the standard bi–linear finite element shape function associatedwith node J (located at x = xJ), TJ is the value of T at node J and a1, . . . , ap

are constants. An approximate weak variational formulation of the problemgiven in equations (3.1)–(3.3) can then be written as

find T ∈ TTT (p) such that ∀ t > t0

∫

ΩρCp

∂T

∂tNI dΩ +

∫

Ωk

∂T

∂xj

∂NI

∂xjdΩ =

∫

ΓN

qNI dΓ +∫

ΩQNI dΩ

(3.5)

for each I = 1, . . . , p. The essential boundary condition at ΓD is specifiedlater when solving the resultant algebraic system of equations. The integralsappearing here can be evaluated by summing individual element contributions,and the compact support of the shape function NI means that the statementmay be re–written as


∑

E∈I

[ ∫

ΩE

ρCpNINJ dΩ]

dTJ

dt+∑

E∈I

[ ∫

ΩE

k∂NI

∂xj

∂NJ

∂xjdΩ

]TJ =

∑

B∈I

[ ∫

ΓB

qNI dΓ]

+∑

E∈I

[ ∫

ΩE

QNI dΩ]

(3.6)

for each I = 1, . . . , p and where the summations just extend over those el-ements E and boundary edges B, ΓB ⊂ ΓN , which contain node I. In thiswork, as ρ, Cp and k are considered constants, a direct numerical integrationover ΩE and ΓB, using a 2 × 2 Gauss quadrature rule [46], can be applied.Performing the summations presented in equation (3.6) and using a compactmatrix notation, problem (3.6) is rewritten as


CdT

dt+ KT = F

(3.7)

Heat Conduction 55

where C is the consistent capacity (“mass”) matrix, K is the conductivitymatrix (generally called the “stiffness” matrix), F is the vector with the in-dependent terms, which arises due to the imposition of thermal loads on thedomain and/or the boundary conditions, and finally T is the vector of thenodal unknowns TJ(t).

3.2.2 Time discretisation

The problem described in (3.7) represents a coupled system of first–order ordi-nary differential equations, which can be further discretised in time to producean algebraic system of equations. As already mentioned in section 2.3.2, anappropriate variational formulation can be identified for the problem (3.7),which can be again solved using finite elements to discretise the “time” do-main. Suppose the time domain (t0,T) is discretised into elements or timeintervals, with nodes numbered 0, 1, . . . , q. The subsets TTT (q) and WWW(q) of thetrial and weighting functions respectively can be defined as

TTT (q) = T (q)(t) | T (q) =q∑

n=1

T nNn(t); T 0 = T0

WWW (q) = W(q)(t) |W(q) = Wn(3.8)

where Nn is a standard linear one–dimensional finite element shape functionassociated with the node placed at time t = tn. The approximate variationalformulation, considering the compact support of the functions Nn, can bewritten as

find T n ∈ TTT (q) such that

∑

I∈(n)

[ ∫

In

(C

dT n

dt+ KT n − F (t)

)Wn dt

]= 0

(3.9)

for n = 0, 1, . . . , q; with the summation extending over the intervals whichcontain the node placed at t = tn and In = (tn, tn+1]. Each choice of theweighting functions Wn leads to a particular time integration formulation.If the independent term F varies smoothly in time, the linear interpolationfunction Nn can also be used for its representation. Adopting Wn = δ(t− tn−θ∆tn+1) for n = 0, 1, . . . , q; represents the point collocation weighted functionat t = tn + θ∆tn+1 for each element (interval) In, section 2.3.2. The problem(3.9) can finally be expressed [45] as the system of linear algebraic equations

K∗T n+1 = F∗

T = T0

at t = t0(3.10)

Heat Conduction 56

with

K∗ = θK +1

∆tn+1C

F∗

= [ K∗ − K ] T n + [ θFn+1

+ (1 − θ)Fn]

(3.11)

where ∆tn+1 = tn+1 − tn represents the length of the time interval In and

with the independent term F and the initial condition T0

known. Equations(3.10) and (3.11) represent a family of two–level time–stepping schemes, whichcan also be derived using the finite difference approach [45]. According to thechoice of the point collocation θ within the interval In, different finite differenceschemes result. Equation (3.10) has to be solved at each time–step if θ 6= 0,with a fully implicit scheme for θ = 1. It reduces to an explicit time integrationformulation if θ = 0 and if the consistent capacity matrix C is diagonalised,or lumped, [45].

The two–level time–stepping schemes are extremely popular, when deal-ing with heat transfer problems, because of their characteristics of easy pro-gramming, small memory requirements and simplicity to alter the time–stepas the solution dictates. These schemes are first–oder accurate in time forall θ ∈ [0, 1] except for θ = 1/2, in which case the method is second–order.For each choice of θ, different properties are present in the resultant scheme[28]. This work will concentrate on the Euler–backward scheme (θ = 1) whichreduces (3.10)–(3.11) to

[ ∆tn+1K + C ] T n+1 = CT n + ∆tn+1Fn+1

(3.12)

The main characteristic of Euler–backward scheme is the fact that it rep-resents the only two–level scheme which is stable and free from oscillationswithout restriction on the time–step adopted [45], with the size of ∆t deter-mined only by accuracy requirements. This makes the Euler–backward schemevery attractive for problems in which a very long time scale is required to reacha stationary condition and also for problems which require very fine spatialdiscretisation. It also possesses enough dissipation to damp higher frequencieswhich dominate the early stages of the transient response in stiff initial valueproblems [11], K/ρCp → 0, or which can lead to the so called aliasing phe-nomena in non–linear problems [22]. For non–linear problems, it representsthe only scheme that retains unconditional stability [23]. The drawbacks ofthe method are the first–order accuracy and the necessity to solve a systemof equations at each time interval. However, the extra cost involved at eachiteration of an implicit method is more than compensated by the larger time–steps which may be taken, and in general implicit methods are more efficient

Heat Conduction 57

for the solution of heat transfer problems, mainly if an automatic time–stepadjustment is available.

It should be noted that in equation (3.9) only an absolute derivative ap-pears in the integrand, and by the fundamental theorem of the integral cal-culus, the integral can be evaluated without requiring any level of continuityat the discrete time levels tn. This allows the freedom to choose a subset ofthe trial functions TTT (q) which includes piecewise discontinuous functions. Theprocedure in which the finite element formulation is used to discretise in time,with such flexibility for TTT (q) and with T n = Wn, is referred to as the discon-tinuous Galerkin method [25]. As in the standard finite element procedure,the trial functions are chosen to be polynomials with low degree g. Employingconstant trial functions, g = 0, on each time interval, the resultant discreteproblem, after using the discontinuous Galerkin procedure, is

[ ∆tn+1K + C ] T n+1 = CT n +∫

In F (t) dt (3.13)

This expression can be recognized as a simple variant of the Euler–backwardscheme (3.12) where the independent term involves the evaluation of the in-

tegral of F (t) over In rather than the value of Fn+1

at tn+1. If F (t) variessmoothly in time, the linear variation of F (t) adopted to derive (3.12) canonce again be used here, which makes (3.13) exactly the same as (3.12).

3.2.3 A–posteriori error estimators

A proper error estimator for parabolic problems must take into account thetwo main errors involved in the discretisation, i.e. the spatial and time dis-cretisation errors. Over the past several years substantial progress has beenmade in the development of a–posteriori error estimators [3, 4, 18, 27, 47, 48],but the majority of this work deals only with the spatial discretisation errorin elliptic problems. Then some well–based mathematical tools can be used tohave an assessment of the qualitative behavior of the solution.

By considering the “right” choice of function space TTT , from a mathemat-ical point of view, and by using the discontinuous Galerkin method, Johnson[25] derived a suitable a–posteriori error analysis for the fully discrete schemegiven in (3.13), which can be written as

maxt∈I

‖ T(t) − T(t) ‖L2 ≤ c[ (

maxn≤q

∆tn+1 ‖ dT(t)/dt ‖∞,In

)ρCp

+

(maxt∈I

h2 ‖ T(t) ‖H2

)K

] (3.14)

Heat Conduction 58

for n = 1, . . . , q, and in which the maximum norm is defined by

‖ v ‖∞,In= sup

z∈In‖ v(z) ‖L2 (3.15)

Here ‖ v ‖L2 is the L2-norm, ‖ v ‖H2 is the norm corresponding to the Hilbertspace H2(Ω) and c is a constant dependent on the spatial and time discretisa-tion. In equation (3.14) the first term on the right hand side bounds the timediscretisation error while the second term bounds the spatial discretisation er-ror. For more detail about this expression, reference [25] is recommend. Theadopted estimate for both portions of the error is now described.

Spatial discrete error (elliptic problem)

The spatial discrete error can be estimated using one of the three distincttypes of available a posteriori error estimators. First, the residual type oferror estimator presented by Babuska and Rheinboldt [3] and later by Gagoet al [18]. Second, an interpolation type estimator derived by Demkowicz et al[15] and also by Johnson [25]. Finally, the projection or variational recoverytype of estimator suggested by Zienkiewicz and Zhu [47]. The third type isadopted as it allows an easy implementation, with no additional data structurerequired. It is also relatively fast to compute, as it requires only the evaluationof the first derivatives of the solution.

The discrete energy norm [46] of the spatial error over an element E,‖ eE ‖s

EN , can be written in terms of the gradient of temperature as

‖ eE ‖sEN=

[ ∫

ΩE

(∇T −∇T

)TD(∇T −∇T

)dΩ]

1/2 (3.16)

and the discrete energy norm of the gradient of temperature over element Eis evaluated as

‖ ∇TE ‖EN=[ ∫

ΩE

(∇T

)TD(∇T

)dΩ]

1/2 (3.17)

where D represents the constitutive matrix, which for the present isotropicmodel is a diagonal matrix equal to k times the identity matrix. If D isreplaced by the identity matrix, the energy norm coincides with the L2-norm,which is adopted in this work.

In the above equations, ∇T denotes the vector of the exact nodal gradi-ents and ∇T denotes the vector of the finite element approximate gradients.Since linear shape functions have been adopted, the finite element gradientsare discontinuous and unrealistic. Oden and Brauchli [34] and later Hinton

Heat Conduction 59

and Campbell [20] have shown a consistent procedure for the calculation ofgradients in the finite element method, which is based on the idea of conju-gate approximations. With only a small increase of computation, a gradientfield ∇T ⋆ consistent with the unknown and more accurate in a least-squaresweighted residual sense is obtained. Oden and Brauchli [34] observe that thefinite element ∇T gradients are mapped completely out of the trial functionssubspace TTT (p) (3.4) of the space TTT containing the actual solution, and define

a conjugate subset TTT (p)∇ according to

TTT (p)∇ = ∇T∇(xj) | ∇T∇ =

p∑

J=1

∇T ⋆J (t)N⋆

J (xj) (3.18)

They suggest a systematic projection procedure for computing ∇T ⋆, which isequivalent to minimize the square of the residual (∇T ⋆ − ∇T ) with respectto the free parameters ∇T ⋆

J (t), i.e.

∂

∂(∇T ⋆J (t))

∫

Ω(∇T ⋆ −∇T )2 dΩ

= 0 (3.19)

This leads to the linear system of equations

M⋆∇T ⋆ = P (3.20)

where

M⋆ =∑

E∈I

[∫

ΩE

N⋆I N⋆

J dΩ]

P =∑

E∈I

[∫

ΩE

N⋆J∇NI dΩ

]TI

(3.21)

for each I = 1, . . . , p , where the summations extend over those elements Ewhich contain node I.

Some remarks are pertinent about this procedure to compute an improvedgradient of the finite element solution:

(1) The improved gradient ∇T ⋆ can be computed explicitly bydiagonalizing the matrix M⋆, i.e. M⋆

L, and successively improvedsolutions can be obtained by iteration [46] using

∇T ⋆(k) = ∇T ⋆(k−1) + (M⋆L)−1

[P − M⋆∇T ⋆(k−1)

](3.22)

In practical applications, it is found that the iterations might becarried out only a fixed number of times , normally 3 or 4, if it

Heat Conduction 60

is judged important. However, is normally unnecessary when thegradient is to be used for adaptive purposes.

(2) The best and most obvious projection, Oden and Brauchli [34],is obtained using N⋆

I = NI , but other projections, even using asimple nodal averaging of the gradients, give good results for thebi–linear elements which are adopted.

(3) This procedure is also called variational recovery and can beused repeatedly to determine an approximation to a higher–orderderivative.

(4) The continuous gradient field which is obtained does not satisfythe variational formulation (3.5). Cantin et al [7] suggest the ap-plication of Loubignac’s iterative scheme to minimize the resultantresidual when necessary.

(5) Based on the well known existence of some exceptional pointswithin the finite elements [5], known in advance for certain el-ements, and where the approximate solution is superconvergente,Zhu and Zienkiewicz [44, 48] recently proposed a more accurate wayto improve the gradient of the approximate solution using patchesof elements to obtain the smoothing. They have demonstratednumerically, that for even shape functions, the procedure couldrepresent a large improvement if compared with the projectionsdescribed above. However, for linear elements the only advantageof the new scheme is the fact that no global algebraic system ofequations has to be solved. This is also true in the described pro-cedure when M⋆

L is adopted.

Observing the best quality of the gradients obtained using the proceduredescribed, Zienkiewicz and Zhu [47] proposed an error estimator ‖ e⋆ ‖s

L2for

a given element E where the exact gradients of the solution ∇T is replacedby the improved gradients ∇T ⋆ in equation (3.16). The global error over thedomain, discretised with m elements, can then be assessed by

‖ e⋆ ‖sL2

=

[m∑

E=1

(‖ e⋆E ‖s

L2)2

]1/2

(3.23)

and the global L2–norm of the temperature gradient is

‖ ∇T ⋆ ‖L2=

[m∑

E=1

(‖ ∇T ⋆E ‖L2)

2

]1/2

(3.24)

with ‖ ∇T ⋆E ‖L2 computed using equation (3.17) with ∇T ⋆ replacing ∇T. The

normalised error in the solution can be estimated as

Heat Conduction 61

η =‖ e⋆ ‖s

L2

‖ ∇T ⋆ ‖L2

(3.25)

The reliability of this error estimator has been established mathematically[1]. It has also been successfully used in adaptive analysis for many types ofelements in different physical problems [46].

Time discrete error (parabolic problem)

To evaluate the fraction of the error corresponding to the time discretisationin (3.14), the estimate proposed by Jonhson et al [17, 25, 27] is adopted. Theirdemonstration is based on the following conditions: (a) the exact solution isbounded; (b) the time–step ratio ∆tn±1/∆tn is also bounded; (c) the ratio∆tn/tn is sufficiently small; (d) the spatial discretisation satisfies a regularitycondition; (e) the initial time–steps are selected separately. The resultant timeerror estimator can be written as

‖ en ‖tL2≤ c∆tn+1 ‖ dT/dt ‖

∞,In≈ c ‖ T n+1 − T n ‖L2 (3.26)

for all n ≤ q, where c is a constant which is solution independent for lin-ear problems. This error estimator is O(∆t), which is compatible with thefirst–order of accuracy of the implicit scheme adopted. The error estimator isnormalised using the present solution according to

ϕ =‖ en ‖t

L2[(‖ en ‖t

L2)2 + (‖ T n ‖L2)

2]1/2

(3.27)

The mathematical validity of this simple error estimator can be found inthe original papers by Jonhson et al [25, 27], where some numerical results forone–dimensional test problems are also presented.

3.2.4 Adaptive spatial and time control algorithms

The error analysis, in which the total error measured with a specific norm isestimated and the relative contribution of the individual elements is indicated,is by itself helpful in the interpretation of the approximate results. However,its potential is better explored when combined with an adaptive process. Anadaptive procedure, in the present context, may be considered a computa-tional algorithm for the construction of a discrete model for a given problemsuch that, ideally, the error of the corresponding approximate solution remains

Heat Conduction 62

within a given tolerance, in a given norm, and such that the number of degreesof freedom and time integrations is minimal.

When dealing with parabolic problems, very small time–steps are in gen-eral necessary during the initial transient to satisfy an accuracy requirement.However, outside this interval, larger time–steps can be used. Therefore, onewould seek an automatic procedure to adapt the time–step according to thesmoothness of the solution, which will reduce the number of systems of equa-tions to be solved and, hence, improve the efficiency.

The standard finite element procedure, presented in sections 3.2.1 and3.2.2, interprets the solution of a parabolic problem as a successive solutionof elliptic problems. In that way, the time and space discretisation can bedecoupled and the error analysis, and also the adaptation procedure, can toa certain extent, be established independently. Most previous adaptive pro-cedures developed to solve parabolic problems define the adaptive procedurebased only on the spatial error. As a loss of accuracy at an intermediate timelevel cannot be recovered in the future solution, it is necessary to choose a verysmall time step to guarantee that this component of the discrete error is negli-gible when compared to the spatial discrete error. In this way, no advantage ofthe progressively smooth nature of many parabolic problems can be achieved.

Mesh refinement

The strategy for the spatial adaptation follow within the so called mesh en-richment procedure and a detailed description for the steady state counterpartcan be seen in references [29, 32]. For the transient problem it can be summa-rized through the following algorithm which is further explained in subsequentremarks.

1. Generate the discrete model;2. Determine the current time–step

(see details given below on the time–step control algorithm);3. Advance the solution over ∆tn+1 by solving the resultant system of

equations;4. Estimate the spatial discretisation error;5. Refine the mesh according to the adopted refinement criteria;6. If at least one element was refined in step 5:

6.1 Reset time over ∆tn+1;6.2 Store the solution at time tn+1 − ∆tn+1;6.3 Go to step 1;

7. If the time of the analysis has elapsed or the solution reaches thesteady–state then stop.

8. Go to step 2, to carry on the time integration;

Heat Conduction 63

Remark S.1 - To start the time integration, the first time–step is definedas

∆t1 =∆test

1000=

1

1000× ρCph

2

4k(1 − 2θ)(3.28)

which represents a time–step of the order of 10−3 times the stability limit ∆test

for an explicit method, in a 2–D model with an uniform mesh [45]. This choiceguarantees that the discrete error (3.14) reduces only to the spatial componentand the algorithm falls back into the steady–state counterpart. The mesh isrefined until the error estimate is below the pre–assigned tolerance. This pre–processor stage produces also a solution which will be used to evaluate thediscrete time error after the first “real” advance in time.

Remark S.2 - In certain applications in which the solution changessmoothly in time, the “optimal” grid determined at previous time level shouldbe a very good approximation to the desired grid at the present time. From anefficiency point of view, it could be interesting that the spatial error is checkedand a possible refinement takes place [30] only after a certain number of time–steps. The number of steps is problem dependent and must be carefully chosento maintain the reliability of the procedure.

Remark S.3 - The algebraic system of equation is solved at each stepusing a preconditioned Conjugate Gradient algorithm [2, 19] implemented inan element–by–element base [23], with the coarser grid solution used as an im-proved initial guess, in a similar way to the so called nested iteration, [6]. Theconjugate gradient method represents one of the most powerful iterative solvertechniques, and the element–by–element implementation has in addition theadvantages of not forming the global matrices, reduced memory requirementsand being suitable for vectorization [9] and for parallelization [42]. The choicefor an iterative scheme is also important due to the complete loss of the bandcharacter of the matrix K∗ as a result of the mesh refinement.

Remark S.4 - An initial structured mesh is used and after the local re-finements, each element is subdivided into four new elements. Non–conformingnodes are introduced in the grid, with the need for an additional data struc-ture. A tree data structure is adopted to keep the necessary information andsome restrictions are imposed at element level to ensure a conforming method,[15, 32].

Remark S.5 - There are several refinement strategies based on the prin-ciple of error equidistribution, with the convergence ratio, the reliabity and thefinal cost of the solution depending on the chosen strategy [29]. A commonproblem of mesh adaptive procedures, based in such principles, is that at-tempts to equidistribute the error tend to concentrate an excessive amount of

Heat Conduction 64

elements in the neighbourhood of a singularity. The presence of other impor-tant features in the solution then might not be detected, compromising thesolution in such regions. This problem is even more serious for hyperbolic oradvection–dominated parabolic problems [13]. A procedure that provides anacceptable solution everywhere in the domain, proposed by Lyra et al [29, 32],consists of a two–stage process to define the refinement:

Stage 1 Define the allowable, or target, error per element, for the wholedomain Ω, as

em = η ×[‖ ∇T ⋆ ‖L2

m

]1/2

(3.29)

with η denoting the prescribed percentage of error. Whenever the normalisederror estimator η, (3.25), in the solution is bigger than η refine all elements forwhich

‖ e⋆E ‖s

L2≥ em and levelE < Maxlevel (3.30)

where Maxlevel represents the maximum number of times a single elementcan be refined and levelE is the current level of refinement of element E. Thisrepresents one of the simplest strategies and reflects the fact that, in mostcases, the interest is not to solve the most expensive model allowable, butrather to achieve the best resolution of the solution associated with a minimummesh size. Furthermore, though the process of mesh enrichment adopted leadsto satisfactory solution with a relative small number of grid points, it mightnot be economical, as the number of trial solutions, i.e. meshes analysed,might be excessive. See reference [29] for some possible improvements on thisstrategy, where for instance an error prediction for the sub–elements is usedto recursively refine the mesh before a subsequent analysis. This leads to lessiterations, and so CPU time, in the adaptive procedure to reach the requiredmesh for the accuracy fulfillment.

Stage 2 Split the computational domain into two subdomains, definedaccording to

Ω• = ∪ΩE|levelE = Maxlevel and ‖ e⋆E ‖s

L2≥ em

Ω = ∪ΩE|levelE < Maxlevel or ‖ e⋆E ‖s

L2< em

(3.31)

where Ω = Ω• ∪ Ω and Ω• ∩ Ω = 0, with Ω• denoting the subdomain in thevicinity of the singularities and Ω the subdomain after purging the subdomainΩ• from Ω. The normalised error in the solution for the non–purged subdomainΩ can be estimated by

Heat Conduction 65

η =

(‖ e⋆ ‖s

L2

)

Ω

(‖ ∇T ⋆ ‖L2)Ω

(3.32)

and the allowable error per element, for the subdomain Ω, is evaluated ac-cording to

em = η ×[

(‖ ∇T ⋆ ‖L2)Ω

m

]1/2

(3.33)

with η representing a pre–assigned percentage of error in Ω, which can bedifferent from η and m is the number of elements in Ω. If η0 > η 0, refine allelements in Ω for which

‖ e⋆E ‖s

L2≥ em and levelE < Maxlevel (3.34)

Remark S.6 - In many heat conduction applications some special fea-tures such as heat sources, boundary heat fluxes, corners and obstacles arelocated at known positions, which are fixed, and the refinement capabilityof the adaptive procedure is necessary to deal with the singularities origi-nated in these special regions. A coarsening capability might not representan advantage for such problems, as it requires a more complex data structureand an associated cost to eliminate elements, nodes and reorganize the datastructure. When these special features move throughout the domain, or a heatconduction–convection problem is considered, the derefinement capability mustbe used as the solution migrates over the domain, with the local refinementfollowing the important features and the previously refined elements returningto a coarser discretisation [16]. Another very effective scheme might be toadopt an adaptive remeshing technique [13, 14, 36].

Time–step control

Many techniques are available to obtain an automatic time–step control, mostof them based on computations of a local error truncation [26], requiring inprinciple the results of a higher order method, which leads to a considerablecost in the process. Alternatively, the discrete error estimator given in equation(3.26) can be limited by a pre–assigned tolerance ϕ during the time integrationand, in order to have an efficient algorithm, the error estimator must be keptclose to this tolerance. To achieve such a target, the time–step must be adaptedand following Johnson et al [25, 27] the time–step size is sought by solving thenon–linear problem

Find ∆tn+1 such that ‖ en ‖tL2

= ϕ with ‖ en ‖tL2

= f(∆tn+1) (3.35)

Heat Conduction 66

This problem can be linearized by estimating the error (3.26) using the back-ward difference on the previous time interval, i.e.

‖ en ‖tL2≈ c ∆tn+1 × ‖ T n − T n−1 ‖L2

∆tn(3.36)

In this way the next time–step ∆tn+1 can be predicted by

∆tn+1 =ϕ

c ‖ T n − T n−1 ‖L2

× ∆tn n ≥ 1 (3.37)

which represents a very simple procedure to adapt the time–step with compar-atively little cost involved in its evaluation. Item 2 in the adaptive algorithmdescribed previously can now be detailed as follow:

2.1 Compute T n;2.2 Estimate the time discretisation normalised error ϕ;2.3 If ϕ is bigger than ϕ:

Then:(a) Reset time over ∆tn;(b) Take ∆tn+1 = ∆tn/2;

Otherwise:(a) Evaluate ∆tn+1 according to equation (3.37);

Remark T.1 - The smallest time–step obtained at the beginning of thealgorithm is kept in memory to be used, as an initial guess, in the case of theexistence of more than one transient in the problem solution.

Remark T.2 - In transient analysis, it is generally thought that the ac-curacy can be improved, indefinitely, as the time–step decreases in magnitude.However, with the finite element approximation, very small time–steps maycauses stability problems or lead to physically unrealistic results. It is known[38] that this behavior is a result of the violation of the discrete maximum prin-ciple [8] when the consistent capacity matrix C is adopted and the time–stepmust satisfy a lower limit. When the capacity matrix is conveniently lumped ordiagonalised, CL, no such limit applies and only the accuracy determines thetime–step magnitude, see appendix A. The special lumping technique givenby Hinton et al [21] is recommended as, for an arbitrary element, it alwaysproduces a positive diagonal matrix, which is required for the unconditionalsatisfaction of the discrete maximum principle. For the isoparametric bi-linearelement adopted here, it is equivalent to the row–sum technique or the use ofa nodal quadrature rule for the numerical integration.

Heat Conduction 67

Remark T.3 - Owing to the discrete nature of the finite element, thediscretisation has to be chosen in such a way that the highest frequency inthe finite spectrum to be predicted accurately can be represented. As arguedin [22], the shortest reasonable wavelength λ′

min represented on a mesh of sizehmin is λ′

min = 2hmin. Based on this physical argument, and on the time that ittakes this fastest decaying component to be reduced to 1/10 of its initial value,Sampaio [12] showed that, for a one–dimensional heat conduction problem,a time–scale approximately equal to third the explicit stability limit shouldbe defined for the problem. This value was adopted for the first time–stepto start the time integration. This represents a very stringent criterion andusing it one expects no, or only very small, reduction in the time–step at thebeginning of the integration in time. The procedure advances with the time–step being adapted, normally increasing, during the interval of the analysis. Ifthe error tolerance is extremely severe, or if the space discretisation is not inagreement with the time discretisation required by the error analysis, then anexcessive reduction is expected in the step. Therefore, together with the spacediscretisation adaptive procedure, where a maximum number of refinementswas defined to limit the minimum mesh spacing, a lower limit in the timediscretisation should be imposed. A lower bound equal to 20% of the explicitstability limit is adopted here. In this way, the error criteria will not be satisfiedonly in an extremely small interval. When a consistent mass matrix is adopted,the lower bound limiter is given by equation (A.8) as described in appendixA.

Remark T.4 - The relation ∆tn+1/∆tn was limited, normally withinthe interval [1.5, 2.0]. This is important due to the approximate nature of theestimate of the error. Actually, this restriction only applies during the startof the transient if the first step is very conservative and when the solutionapproaches the steady–state, when the error decreases very rapidly.

Remark T.5 - No study was done to investigate the constant c involvedin the time error estimator, as the main objective was to use this error as anindicator to determine the suitable time–step. The value of such a constantcould be important if a sharper prediction of the error estimation is desiredfor a special problem.

3.3 Numerical Applications

From an efficiency point of view, it is always advantageous to begin an adap-tive approach with a suitable mesh, provided by a pre–analysis based on theanalyst’s knowledge. However, in order to stress importance of the mesh adap-tive as a pre–processor, very coarse initial discretisations are adopted in thefollowing model applications. The simple applications presented focus on the

Heat Conduction 68

time–step adaptive procedure. The performance of the mesh refinement pro-cedure for both steady–state and transient analysis can be seen in references[30, 31], in which the proposed procedure is applied to analyse groundwaterflow, including a radiation type (leakage) term in the governing equation andin the boundary conditions.

3.3.1 Model problem I

The problem geometry, initial discrete model, material properties, boundaryand initial conditions are presented in figure 3.1. The tolerances adopted forboth the space discrete error η and the time discrete error ϕ is 10%. Themaximum level of refinement, Maxlevel, is six and a bound of 3/2 is used inthe relation ∆tn+1/∆tn.

X2

X1

ΓN

ΓD 1

4

k = 1.0

T = 1.0q = 0.0

ρCp = 1.0

T = 0.00

Figure 3.1: Physical characteristics, geometry and initial mesh for model prob-lem I.

The final mesh which is expected for this specific problem is the initialmesh, but as the code does not allow coarsening, the mesh after the pre–processor stage in the algorithm is also the mesh used throughout the analysis,see figure 3.2. This mesh produces the transient solutions presented in figure3.3, which are free from any sort of oscillations and are in good agreement withthe analytical solution presented in reference [11].

The time–step adaptation history and the time discrete error evolutionduring the time integration, both in the L2–norm and ∞–norm are presentedin figure 3.4. The growth of the time–step with the smoothness of the solutionis clearly observed and the adopted strategy limits the L2–norm below 0.1 asdesired.

The same problem was analysed using the mesh presented in figure 3.2and a fixed time–step equal to the smallest value obtained during the adaptiveprocedure, ∆tmin = 2.44 × 10−4. A detail of the discrete time error evolution

Heat Conduction 69

Figure 3.2: Adapted mesh utilized in the analysis of the first model problem I.

(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

t = 0.01t = 0.12t = 1.02t = 10.5t = 97.5

T

X1

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

X1 = 1.0X1 = 2.0X1 = 4.0

T

t

Figure 3.3: Numerical solution for model problem I: (a) temperature distribu-tion at sample times; (b) temperature history at sample points.

at the beginning of the transient is presented in figure 3.5(a) where it can beseen that the error is kept below and close to the adopted tolerance of 0.1,which is, in some sense, optimal. The use of the fixed time–step ∆tmin, thatsatisfies the tolerance at the initial step, becomes quickly too conservative,emphasizing the importance and efficiency of controlling the time–step. Forboth analyses, the spatial error has similar values, with the global estimatorη ≈ 5.9%. However the estimator in domain Ω presents some difference withη ≈ 3.22% with time–step adaptation and η ≈ 0.64% with a fixed step. Thisrepresents an interesting result, showing the interdependence of the two partsof the discrete error. To cover the time of the analysis presented in figure 3.5,only 18 steps was necessary with the time–step control against 130 withoutadaptation, reducing the number of algebraic system of equations to be solvedmore than seven times. The solution along the domain [0, 2] in the x1 directionis given in figure 3.5(b) where no loss of accuracy in the solution is seen whenthe adaptation is used.

It was observed that the limitation on the time–step relation ∆tn+1/∆tn

is important to preserve the quality of the solution during the whole analysis,

Heat Conduction 70

(a) (b)

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

t

t∆

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100

L2-norm00-normϕ

t

Figure 3.4: Time evolution for model problem I: (a) time–step; (b) time discreteerror estimate.

(a) (b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.005 0.01 0.015 0.02 0.025 0.03

Fixed Adapt

t

ϕ

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Fixed AdaptT

1X

Figure 3.5: Fixed and adaptive time–step during the beginning of a transientfor model problem I. (a) Time discrete error history; (b) detail of the temper-ature distribution at time 3.1 × 10−2.

but it also compromises the efficiency of the procedure if a value very close tothe unity is adopted. If the ∞–norm is used instead of L2–norm, one couldexpect more freedom in that choice but it seems to be a overly conservativeoption.

3.3.2 Model problem II

The second model problem consists of an “infinite” domain with a heat source.The initial finite discrete model adopted in the analysis and the physical char-acteristics of the problem are presented in figure 3.6. The domain is truncated

Heat Conduction 71

at a distance r = 250.0. The tolerance for the discrete error estimators η andϕ is η = ϕ = 20% and the maximum level of refinement is again 6 with a limiton the time–step relation of two.

2x

1x

K = 1.0

Q = 1.6

ρCp = 10 -8

T = 1.0

T = 0o

Q

ΓD

Figure 3.6: Physical characteristics, geometry and initial mesh for model prob-lem II.

The final meshes obtained using the transient and also the steady–stateprocedures are shown in figure 3.7. The global spatial error estimator η atsteady–state for both strategies are 37.33% and 37.59% and the errors over thedomain Ω are 6.99% and 4.76%, respectively. The criteria described to dealwith singularities (Remark S.5 in section 3.2.4) was automatically activatedin both cases with some extra refinements taking place away from the heatsource. In the present application, η was taken to be the same as η.

A sketch of the temperature distribution at certain times in the regionclose to the heat source is depicted in figure 3.8. Further detail of the temper-ature distribution can be seen in figure 3.9, where a semi–analytical solutionusing Bessel functions [10] is presented for comparison. The numerical solutionis in reasonable agreement with the semi–analytical solution. Better resultsare expected if a tighter tolerance for the error is adopted.

The evolution of the spatial and time discrete error is presented in figure3.10. Figure 3.10(a) shows that the maximum number of refinements allowedis incompatible with the tolerance required for the spatial error and also showsthat the majority of the global error (≈ 35%) refers to the subdomain Ω•

Heat Conduction 72

(a)

(b)

Figure 3.7: Final meshes for model problem II. (a) Transient analysis; (b)steady–state analysis.

around the strong singularity existent in the solution. Despite the fact thatthe global error η is higher than the pre–assigned tolerance, the error outsidethe neighborhood of the singularity is kept well below the tolerance η ≈ 5%and the solution over the subdomain Ω is reliable, as the singularity effectno longer interferes in this region. The time error evolution is shown in figure3.10(b), which also shows the presence of a singularity in time due to the

Heat Conduction 73

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-60 -40 -20 0 20 40 60

t = 1.0E-6t = 1.0E-5t = 1.0E-4t = 1.0E-1S-S

T

x1

Figure 3.8: Detail close to the source of the temperature distribution througha diameter at sample times, for model problem II.

(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 50 100 150 200 250

Semi-Anal. Adapt/FEM T

1x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150 200 250

Semi-Anal. Adapt/FEM T

x1

Figure 3.9: Comparison between “analytical” and numerical temperature dis-tribution through a diameter, for model problem II. (a) At time 1.0×10−4; (b)at the steady–state.

incompatibility between the initial condition and the heat source activatedinstantaneously. If no constraint is adopted for the smallest time–step obtainedby the adaptive strategy it leads to ∆tmin = 2.33× 10−12, which is well belowthe stability limit for explicit time integration ∆test = 9.54 × 10−9. Thisjustifies the use of a lower limit, as suggested in Remark T.3.

The time–step evolution exhibits a similar behavior as presented for theprevious application, 3.4(a) and is not shown. Using the mesh given in figure3.7(a), and a fixed time–step equal to the explicit stability limit ∆test = 9.54×

Heat Conduction 74

(a) (b)

0

10

20

30

40

50

60

70

0 0.001

Global Purged

η

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.001

L2-normoo-norm

t

ϕ

Figure 3.10: Error estimators history for model problem II: (a) Spatial; (b)time.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1e-06

FixedAdapt

ϕ

t

Figure 3.11: Time discrete error history using fixed and adaptive time–stepduring the beginning of a transient for model problem II.

10−9, the problem was reanalysed in order to compare with the adaptive time–step solution. In figure 3.11, the time error evolution during the initial phaseof the transient is shown, and once again it can be seen how conservative andinefficient is the use of a fixed time–step, which leads to almost no error inthe time discretisation far below the pre–established accuracy. On the otherhand, the adaptation of the time–step keeps the error just a little lower thanthe desired tolerance. It must be noted that to reach the time level t = 0.1,which already characterizes the steady–state, only 83 steps is required with theadaptive procedure while approximately 1×107 time steps would be required ifa fixed step is adopted. Obviously much bigger time steps are normally used asan implicit time integration allows, but with no guarantee of accuracy for thetransient solution. Furthermore, if only the steady–state solution is required,the steady–state heat conduction equation can be solved directly with no time

Heat Conduction 75

integration involved.

(a) (b)

0

0.1

0.2

0.3

0 5 10 15 20

FixedAdapt

T

x1

0

0.02

0.04

0 1e-06 2e-06 3e-06

FixedAdapt

T

t

Figure 3.12: Fixed and adaptive time–step solutions during the beginning of atransient for model problem II. (a) Detail of temperature distribution at time3.6 × 10−6; (b) temperature evolution at x1 = 31.25.

At the very beginning of the time integration, the time–step was reducedand later it increases consistently with the smoothness of the solution, withthe step size growth being limited by the relation between time–steps whenthe solution approaches the steady–state. A reasonable agreement betweenthe solutions using fixed and adaptive time–step is achieved, figure 3.12. Thedifferences are probably due to the very crude tolerance adopted for the error.

3.4 Concluding Remarks

The structure of the heat conduction equation is identical to that governing anumber of field problems and, by operating on the appropriate field variables,the procedures developed to analyse heat conduction can also be directly em-ployed for these other problems, such as seepage, inviscid incompressible irro-tational flow, torsion, electric conduction, etc.

Although the analyses presented for simple model problems are not con-clusive, they give some insight into the main characteristics of the proposedtransient adaptive procedure, in particular with respect to the time–step con-trol. The strong stability and oscillation–free behavior of the Euler–backwardscheme together with an accuracy control and an adaptive time–step size defi-nition seems to be the best choice to deal with the class of problems analysed.In order to have a fully adaptive scheme, it is necessary to adapt the stepalso in space. Johnson et al [27] have already presented some ideas in that

Heat Conduction 76

area. The variation of the step throughout the domain represents the im-plicit counterpart of the local time–stepping approach extensively used withexplicit time integration, [33, 39, 43]. Another possibility consists in subdivid-ing the domain and to employ an implicit–explicit algorithm [24, 40] where,in the implicit region, the time–step can be adapted as proposed in this work.This procedure could be very efficient when the subdomains are determinedautomatically and the solution of coupled problems is attempted. The maindifficulty is the definition of an appropriate procedure to interchange informa-tion across the boundaries of the two subdomains, which requires a specificdata structure.

To ensure more flexibility and to allow the solution of more complex prob-lems, in which for instance the heat source moves or coupled problems [14, 41],the derefinement capability is essential, and the unstructured remeshing tech-nique presents as a good alternative approach to be combined with the ideasdescribed in this chapter. The promise of the very high convergence ratereached when p–adaptive refinement is adopted in an “optimally” subdividedmesh [37], e.g. h–p or p–remeshing strategies, and the confirmed good per-formance of such approaches [16, 35] represent yet another possible choice forthe spatial adaptivity.

The use of the technique to eliminate the degrees of freedom in which thesolution has already converged, suggested by Devloo [16] in a different context,seems to be doubly promising. Firstly, it reduces the number of equations tobe solved, and secondly only the region where the solution is still in evolutioncontributes for the error evaluation resulting in a more accurate prediction ofthe necessary time step. Consequently, good results are expected for the wholedomain when the solution is close to steady–state, even if a bigger value of thelimit on time step relation ∆tn+1/∆tn is adopted.

The mesh adaptive procedure combined with a time–step control de-scribed offers computational savings, without compromising the accuracy ofthe results, when compared with the solely mesh adaptive procedure normallyused. To summarize, adaptive procedures save time in data preparation ormesh input data and provide more efficiency and reliability of the analysis,mainly when the problem has little predictability or when the analyst has noprevious experience on similar problems.

Bibliography

[1] M. AINSWORTH, J.Z. ZHU, A.W. CRAIG, and O.C. ZIENKIEWICZ.Analysis of the Zienkiewicz–Zhu A–Posteriori Error Estimator in the Fi-nite Element Method. Int. J. Num. Meth. Engng., 28:2161–2174, 1989.

[2] O. AXELSSON and V.A. BARKER. Finite Element Solution of BoundaryValue Problems: Theory and Computation. Academic Press Inc., 1984.

[3] I. BABUSKA and W.C. RHEIHNBOLDT. A Posteriori Error Estimatesfor the Finite Element Method. Int. J. Num. Meth. Engng., 12:1597–1615,1978.

[4] I. BABUSKA, O.C. ZIENKIEWICZ, J.P.R. GAGO, and A. OLIVEIRA,editors. Accuracy Estimates and Adaptive Refinements in Finite ElementComputations. John Wiley & Sons, 1986.

[5] J. BARLOW. Optimal Stress Locations in Finite Element Models. Int.J. Num. Meth. Engng., 10:243–251, 1976.

[6] L.W. BRIGGS. A Multigrid Tutorial. Society for Industrial and AppliedMathematics, Philadelphia, Pennsylvania, 1987.

[7] G. CANTIN, G. LOUBIGNAC, and G. TOUZOT. An Iterative Algorithmto Build Continuous Stress and Displacement Solutions. Int. J. Num.Meth. Engng., 12:1493–1506, 1978.

[8] P.G. CIARLET and R.A. RAVIART. Maximum Principle and UniformConvergence for Finite Element Method. Comp. Methods Appl. Mech.Engng., 2:17–31, 1973.

[9] A.L.G.A. COUTINHO, J.L.D. ALVES, N.F.F. EBECKEN, and L.M.TROINA. Conjugate Gradient Solution of Finite Element Equations onthe IBM 3090 Vector Computer Utilising Polynomial Preconditionings.Comp. Meth. Appl. Mech. Engng., 84:129–145, 1990.

[10] E. CUSTODIO and M.R. LLAMAS. Hidrologia Subterranea. EdicionesOmega, 1983. In spanish.

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Heat Conduction 78

[11] F. DAMJANIC and D.R.J. OWEN. Practical Considerations for ThermalTransient Finite Element Analysis Using Isoparametric Elements. NuclearEngineering Design, 69:109–126, 1982.

[12] P.A.B. DE SAMPAIO. A Petrov–Galerkin Formulation for the Incom-pressible Navier–Stokes Equations using Equal Order Interpolation forVelocity and Pressure. Int. J. Num. Meth. Engng., 31:1135–1149, 1991.

[13] P.A.B. DE SAMPAIO, P.R.M. LYRA, K. MORGAN, and N.P. WEATH-ERILL. Petrov–Galerkin Solutions of Incompressible Navier–Stokes Equa-tions in Primitive Variables with Adaptive Remeshing. Comp. Meth. Appl.Mech. Engng., 106:143–178, 1993.

[14] P. DECHAUMPHAI and K. MORGAN. Transient Thermal–StructuralAnalysis Using Adaptive Unstructured Remeshing and Mesh Movement.In Proc. of the First Thermal Structures Conference, Virginia, 1990.

[15] L. DEMKOWICZ, P.R.B. DEVLOO, and J.T. ODEN. On an h–TypeMesh Refinement Strategy Based on a Minimization of the InterpolationError. Comp. Methods Appl. Mech. Engng., 3:67–89, 1985.

[16] P.R.B. DEVLOO. An H–P Adaptive Finite Element Method for SteadyCompressible Flow. PhD thesis, University of Texas at Austin, 1987.

[17] K. ERIKSSON and C. JOHNSON. Error Estimates and Automatic TimeStep Control for Nonlinear Parabolic Problem, I. SIAM J. of NumericalAnalysis, 24:12–23, 1987.

[18] J.P.R. GAGO, D.W. KELLY, O.C. ZIENKIEWICZ, and I. BABUSKA.A Posteriori Error Analysis and Adaptive Processes in the Finite ElementMethod – part 1. Int. J. Num. Meth. Engng., 19:1621–1656, 1983.

[19] G.H. GOLUB and C.F. VAN LOAN. Matrix Computations. The JohnsHopkins University Press, 1983.

[20] E. HINTON and J. CAMPBELL. Local and Global Smoothing of Dis-continuous Finite Element Functions Using a Least Square Method. Int.J. Num. Meth. Engng., 8:461–480, 1974.

[21] E. HINTON, T. ROCK, and O.C. ZIENKIEWICZ. A Note on MassLumping and Related Processes in the Finite Element Method. Earth-quake Engineering and Structural Dynamics, 4:245–249, 1976.


Heat Conduction 79

[23] T.J.R. HUGHES. The Finite Element Method: Linear Static and Dy-namic Finite Element Analysis. Prentice–Hall, Inc., 1987.

[24] T.J.R. HUGHES and W.K. LIU. Implicit–Explicit Finite Element inTransient Analysis: Implementation and Numerical Examples. J. Appl.Mech., 2:375–378, 1978.


[26] C. JOHNSON. Estimates and Adaptive Time–Step Control for a Classof One–Step Methods for Stiff Ordinary Differential Equations. SIAM J.of Numerical Analysis, 25:908–926, 1988.

[27] C. JOHNSON, Y.Y. NIE, and V. THOMEE. An a Posteriori Error Esti-mate and Automatic Time Step Control for a Backward Discretization ofa Parabolic Problem. SIAM J. of Numerical Analysis, 27:277–291, 1990.

[28] P.R.M. LYRA. Finite Element Analysis of Parabolic Problems: CombinedInfluence of Adaptive Mesh Refinement and Automatic Time Step Con-trol. J. Braz. Soc. Mechanical Sciences, 15:172–198, 1993. Also publishedinternally at University College of Swansea Report CR/712/92 (1992).

[29] P.R.M. LYRA, J.L.D. ALVES, A.L.G.A. COUTINHO, L. LANDAU, andP.R.B. DEVLOO. Comparison of Local Mesh Refinement Strategies forthe H–Version of Finite Element Method. In Proc. of the X Iberian–Latin–American Congress on Computational Methods in Engineering, volume 2,pages A/595–A/610, Porto/Potugal, 1989.

[30] P.R.M. LYRA and J.J.S.P. CABRAL. Transient Mesh Refinement Pro-cedures for Finite Element Analysis of groundwater flow. In Proc. of theConference on Computer Methods in Water Resources II – GroundwaterModeling and Pressure Flow, volume 1, Rabat/Marocco, 1991.

[31] P.R.M. LYRA, J.J.S.P. CABRAL, and J.A. CIRILO. H–Adaptive MeshRefinement Procedures of Finite Element Methods Applied in the Studyof Ground Water Flow. In Proc. of the VIII Brasilian Symposion onWater Resources, Foz do Iguacu/Brasil, 1989. In portuguese.

[32] P.R.M. LYRA, A.L.G.A. COUTINHO, J.L.D. ALVES, and L. LANDAU.The H–Version Auto–Adaptive Refinement Strategy in the Finite Ele-ment Method. In Proc. of the IX Iberian–Latin–American Congress onComputational Methods in Engineering, Cordoba–Argentine, 1988. In por-tuguese.

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[34] J.T. ODEN and H.J. BRAUCHLI. On the Calculation of Consisten StressDistribution in Finite Element Applications. Int. J. Num. Meth. Engng.,3:317–325, 1971.

[35] J.T. ODEN, L. DEMKOWICZ, T. WESTERMANN, and W. RACHOW-ICZ. Toward a Universal H–P Adaptive Finite Element Strategy. Comp.Methods Appl. Mech. Engng., 77:113–180, 1989.


[37] E. RANK. An Adaptive H–P Version in Finite Element Method. InProc. of NUMETA – Numerical Techniques for Engineering Analysis andDesign, 1987.

[38] E. RANK, C. KATZ, and H. WERNER. On the Importance of theDiscrete Maximum Principle in Transient Analysis Using Finite ElementMethod. Int. J. Num. Meth. Engng., 19:1771–1782, 1983.

[39] P. SMOLINSKI. A Variable Multi–Step Method for Transient Heat Con-duction. Comp. Methods Appl. Mech. Engng., 86:61–71, 1991.

[40] T.E. TEZDUYAR and J. LIOU. Adaptive Implicit–Explicit Finite Ele-ment Algorithms for Fluid Mechanics Problems. Comp. Methods Appl.Mech. Engng., 78:165–179, 1990.

[41] A.R. WIETING, P. DECHAUMPAI, K.S. BEY, E.A. THORNTON, andK. MORGAN. Application of Integrated Fluid–Thermal–Structural Anal-ysis Methods. Thin–Walled Structures, 11:1–23, 1991.

[42] R.B. WILLMERSDORF. Distributed Algorithms for Mesh Generationand Incompressible Fluid Flow. PhD thesis, University College of Swansea,1993.

[43] X.D. ZHANG, J.-Y. TREPANIER, M. REGGIO, and R. CAMARERO.A New Local Time–Stepping Approach for the Unsteady Euler Equations.Technical Report 94–0525, AIAA Paper, 1994.

[44] J.Z. ZHU and O.C. ZIENKIEWICZ. Superconvergente Recovery Tech-nique and A Posteriori Error Estimators. Int. J. Num. Meth. Engng.,30:1321–1339, 1990.

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[45] O.C. ZIENKIEWICZ and K. MORGAN. Finite Elements and Approxi-mation. John Wiley & Sons, 1983.



[48] O.C. ZIENKIEWICZ and J.Z. ZHU. Superconvergent Recovery Tech-niques and A Posteriori Error Estimation in F.E.M. – Part 1 A Gen-eral Superconvergent Recovery Technique. Int. J. Num. Meth. Engng.,33:1331–1382, 1992.

Heat Conduction 82

Chapter 4

Adaptive Scheme for TransientIncompressible Viscous FlowComputation

4.1 Introduction†

One of the most promising and challenging ideas in modern computationalfluid dynamics involves the use of adaptive strategies, wherein the structureof the flow solver and of the discrete model change dynamically during theanalysis in an attempt to optimize the solution procedure, in the sense thatthe best solution is achieved for the least computational effort. The assessmentof the results quality and adaptive criteria is either based on the estimate ofthe error or, at least, by the indication of the evolution of the flow featuresthroughout the computational domain.

In this chapter, an adaptive scheme for both steady and unsteady incom-pressible fluid flow problems is developed. The strategy considered involvesa complete redefinition of the mesh, the so called adaptive remeshing. Thea posteriori error estimator discussed in chapter 3, but now based on the ve-locity gradients, is used to judge the quality of the results and to provide thenecessary information to generate a new improved mesh. Technical problemsrelated to the presence of singularities in the solution, the accumulation ofinterpolation error inherent to a transient remeshing strategy and some otherissues are also addressed.

Among the available combination of dependent variables, for the solu-tion of incompressible fluid phenomena, the methods which adopt primitivevariables are the most attractive, primarily because these variables are those

†Research done in collaboration with P.A.B. de Sampaio.

Incompressible Viscous Flow 84

that are generally measured in a laboratory. Also, the lower–order of theequations provides a relatively straightforward imposition of boundary condi-tions and extension for 3–D simulations. Two major difficulties arise whenattempting to use standard finite element formulations to handle the incom-pressible Navier–Stokes equations when written in primitive variables. Thefirst difficulty arises due to the presence of the convective terms, which ren-der the equations non–self–adjoint. The second difficulty refers to the needto choose compatible interpolations for velocity and pressure, as dictated bythe Babuska–Brezzi stability condition [14, 17]. The first difficulty mentionedabove led to the adoption of the finite element mixed formulations as opposedto standard Galerkin approach, but the second difficulty persists. Although nu-merous convergent combinations of velocity and pressure elements have beendeveloped, the inconvenience involved in their implementation is enhancedwhen adaptive procedures are attempted and/or 3–D extensions are devised.These deficiencies have motivated the development, over the last decade or so,of the generalised Galerkin methods [1, 8, 9, 12, 14, 16, 17, 19, 24]. Thesemethods, either directly or indirectly, make use of the characteristics to ob-tain time–accurate approximate solutions for transport phenomena in fluidmechanics and heat transfer.

Sampaio [4, 6] has shown how the use of the least–squares method, in asimilar way to that employed by Jiang and Carey [16] for hyperbolic problems,and a Galerkin discretisation in the time discretized equations lead to formssimilar to those of SUPG methods [1, 14]. One distinctive feature of suchan approach is the derivation of a Poisson equation for pressure through thecombination of the continuity equation and the minimisation of the squaredresiduals of the momentum balances. The resulting weighting functions dependon the time–step used to advance the solution, with the time–step selected ac-cording to the time–scales of the physical processes which are representableon the mesh. In this case, a correct amount of streamline upwinding is intro-duced and stable pressure solutions are attained for any interpolation of theflow variables.

The possibility to use low– and equal–order elements makes such formu-lations very attractive for implementation in conjunction with adaptive meshrefinement. The formulations, and some general issues involved in their imple-mentation, will be addressed in this chapter.

4.2 Numerical Formulation

The equations governing the motion of an incompressible Newtonian fluid withconstant viscosity can be expressed, see section 2.5.2, in non–conservative form,as a function of the primitive variables UT = [ p, u1, u2 ], as


∇ · u =∂ui

∂xi

= 0

ρ∂ui

∂t+ ρu · ∇ui −

µ

Re∞∇2ui +

∂p

∂xi= ρfi

in Ω × I (4.1)

with i = 1, 2 for 2–D simulations and where u = (u1, u2) denotes the velocityvector relative to cartesian coordinates. The detailed derivation of the flowsolution formulations can be found in [4, 5, 6] and only the main steps arepresented next. Initially, the momentum equations are discretised in timeusing a θ–averaged finite difference scheme. As the Crank–Nicolson scheme,θ = 1/2, gives second–order time accuracy it was adopted, and leads to

Fi − bn+1/2i = 0 (4.2)

where bi = ρfi denote the body forces, and

Fi =ρ

∆t(un+1

i − uni ) + ρun · ∇u

n+1/2i − µ

Re∞∇2u

n+1/2i +

∂pn+1/2

∂xi

(4.3)

The superscript n + 1/2 refers to the arithmetic average of the variable valuesat time–levels n and n+1, and a suitable linearisation for the convective termhave been achieved by the use of the approximation un+1/2 ≈ un. Supposethat the spatial domain Ω has been discretised using linear triangular elements.An approximate solution which belongs to the subset of the trial functionsTTT (p) is sought, as discussed in section 2.3.2. The subset of trial functions hasdimension p, equal to the number of discrete nodes, and is defined by

TTT (p) = U(xj , t) | U =p∑

J=1

UJ(t)NJ(xj); UJ(t0) = U0(xj) = U0

J (4.4)

with U0

denoting the initial condition in Ω at time t = 0, and the same trial orshape functions NJ have been employed for all components of U. Inserting theapproximate solution U represented in (4.4) into expressions (4.2)–(4.3) leadsto the approximated quantities Fi and a least–squares discrete formulation ofthe problem given in equation (4.1) can be stated as

find U ∈ TTT (p) such that Π(U) ≤ Π(V) ∀ V ∈ TTT (p)

and ∀ t > t0 with: Π =2∑

i=1

m∑

E=1

∫

ΩE

[Fi − b

n+1/2i

]2dΩ

(4.5)


Here it should be observed that all terms are defined on the element interiors,where the shape functions are of class CCC∞, thus avoiding the CCC1 inter–elementcontinuity requirements of standard least–squares formulations when used tosolve second–order partial differential equations. The solution of the equivalentproblem described in (4.5) is achieved by minimizing the functional Π withrespect to the free parameter Un+1

J , i.e.

find U ∈ TTT (p) such that ∀ t > t0

∂Π

∂Un+1J

=2∑

i=1

m∑

E=1

∫

ΩE

2[Fi − b

n+1/2i

] ∂Fi

∂Un+1J

dΩ = 0(4.6)

This expression can be re–written as

find U ∈ TTT (p) such that ∀ t > t0

2∑

i=1

m∑

E=1

∫

ΩE

[Fi − b

n+1/2i

](NI + WI) dΩ = 0

(4.7)

which can also be regarded as a Petrov–Galerkin weighted approximation ofthe momenta balances, where

WI =∆t

2un · ∇NI +

µ∆t

2ρRe∞∇2NI (4.8)

Note that the second–order term in WI vanishes and that the resulting weight-ing function is discontinuous across element boundaries for the adopted CCC0

shape function NI . The parameter ∆t/2 acts to bias the integral in equa-tion (4.7) in favour of the upwind term of the trial function, i.e. first term of(4.8), and so upwinding is incorporated into finite element framework in thestreamline direction. The resultant weighting functions NI +WI have an equiv-alent structure to those employed in the standard SUPG [14], and match thepresent formulation, for steady–state computations, if the time–step is chosenaccording to

∆t = α0h/|u|α0 = coth((Re)E/2) − 2/(Re)E with (Re)E = ρ|u|h/µ

(4.9)

where h is the element size and (Re)E is the element Reynolds number. Nodallyexact steady–state solution is obtained for the 1–D convection–diffusion equa-tion when (4.9) is used [37]. Sampaio [5] presented some heuristic arguments,concerning the time–scales of convection–diffusion processes which are rep-resentable in a given mesh, to justify the use of the expression (4.9) for a


multidimensional incompressible Navier–Stokes algorithm. The computationof ∆t according to (4.9) is also just within the stability limit necessary for anequivalent explicit formulation.

A term which refers to the viscous flux boundary conditions and their com-patibility across element interfaces, ΓE, must be added to the Petrov–Galerkinmethod given in equation (4.7) in order to have a well–posed formulation [4].Here this is enforced, in a weak form, according to

2∑

i=1

m∑

E=1

∫

ΩE

[Fi − b

n+1/2i

](NI + WI) dΩ

+m∑

E=1

∫

ΓE

[µ∇u

n+1/2i · n − µ∇u

n+1/2i · n

]NI dΓ = 0

(4.10)

Assuming the boundary conditions

u = u on ΓD × I

µ∇ui · n = 0 on ΓN × I

with

ΓD ∪ ΓN = Γ

ΓD ∩ ΓN = 0

(4.11)

and the compatibility conditions for the exact viscous–flux

[|µ∇ui · n|] = 0 on ΓE × I (4.12)

the approximate weak form of the problem (4.1), for the momentum equations,becomes

∫

Ω

ρ

∆t

(NI +

∆t

2un · ∇NI

)(un+1

i +∆t

2un · ∇un+1

i

)dΩ

+∫

Ω

µ

2Re∞∇NI · ∇un+1

i dΩ = −∫

Ω

µ

2Re∞∇NI · ∇un

i dΩ

+∫

Ω

ρ

∆t

(NI +

∆t

2un · ∇NI

)(un

i − ∆t

2un · ∇un

i

)dΩ

+∫

Ω

(NI +

∆t

2un · ∇NI

)(−∂pn+1/2

∂xi

+ bn+1/2i

)dΩ

(4.13)

An alternative formulation can be devised if a Taylor expansion in time[23] is considered, which up to second–order can be written as


un+1i = un

i + ∆t∂un

i

∂t+

∆t2

2

∂2un+θi

∂t2(4.14)

with 0 ≤ θ ≤ 1. Assuming θ = 0, a time discretisation of (4.1) can beexpressed, after some manipulation, as

ρ

(un+1

i − uni

∆t

)+ ρun · ∇un

i − µ

Re∞∇2un

i +∂pn+1/2

∂xi− b

n+1/2i

−∆t

2∇ ·

[un

(ρun · ∇un

i − µ

Re∞∇2un

i +∂pn+1/2

∂xi

− bn+1/2i

)]= 0

(4.15)

A spatial discretisation is performed using the Galerkin finite elementprocedure, which is justified due to the equivalence with the characteristic–Galerkin method [27] for the 1–D convection problem, where considerations ofcharacteristics, along which the equations have self–adjoint nature, are used.The approximate weak variational formulation of the problem is then

∫

Ω

ρ

∆tNI u

n+1i dΩ =

∫

Ω

ρ

∆tNI u

ni dΩ −

∫

Ωρ(NI +

∆t

2un · ∇NI

)un · ∇un

i dΩ

−∫

Ω

µ

Re∞∇NI∇un

i dΩ +∫

Γρ∆t

2NIu

n · nun · ∇uni dΓ

−∫

Ωρ(NI +

∆t

2un · ∇NI

)(∂pn+1/2

∂xi− b

n+1/2i

)dΩ

+∫

Γ

∆t

2NIu

n · n(

∂pn+1/2

∂xi

− bn+1/2i

)dΩ

(4.16)

This scheme represents the one–step explicit Taylor–Galerkin method origi-nally presented by Donea [8] to deal with convective transport problems andfurther extended and applied to compressible flows and shallow water prob-lems [19, 27]. Sampaio [4] has shown the complete equivalence of these twodistinct approaches, i.e. (4.13) and (4.16), when θ = 1/2 and ∆t is computedusing (4.9), for the 1–D steady–state convection–diffusion equation, justifyingthe adoption (4.9) in both schemes.

The continuity equation has not yet been considered and a discretisationfollowing the standard mixed formulation is dismissed, as the same interpo-lation for all variables has been used and this would imply the violation ofBabuska–Brezzi condition [14, 17]. A key point of Sampaio’s formulations is


the utilization of the incompressibility condition, (4.1)(a), to modify equation(4.6) when the sum of the squared residuals is minimized with respect to the

pressure parameters. By approximating un+1/2i ≈ un

i in the convective termand after some manipulation his approach leads to

∫

Ω∇NI · ∇pn+1/2 dΩ =

∫

Ω

∂NI

∂xi

(bn+1/2i − ρun · ∇un

i

)dΩ

−∫

Ω

ρ

∆tNI∇ · un dΩ −

∫

ΓvD

ρ

∆tNI(u

n+1 − un) dΓ

(4.17)

which represents an elliptic–type equation. The solution of this equation issought over the domain Ω for all t ≥ t0, subject to the boundary conditions

p = p on ΓpD × I

u · n = u on ΓvD × I

with

ΓpD ∪ Γv

D = Γ

ΓpD ∩ Γv

D = 0

(4.18)

where, if ΓpD ≡ 0, at least one pressure reference value must be prescribed to

determine a unique pressure field.

The integrals in equations (4.13), (4.16) and (4.17) can be evaluated bysumming individual contributions and, by adopting a compact matrix notation,the two alternative formulations discussed can be summarized as

Kppn+1/2 = H1un1 + H2u

n2 + Bp

Kvun+1i = Jun

i + Dpn+1/2 + Bv i = 1, 2

(4.19)

for the pressure–continuity/least–squares approach or

Kppn+1/2 = H1un1 + H2u

n2 + Bp

MLun+1i = J′un

i + D′pn+1/2 + B′v i = 1, 2

(4.20)

for the pressure–continuity/Taylor–Galerkin procedure. In these equations,ui and p denote the vectors formed by the nodal values of the component iof the velocity and of the pressure, respectively. In (4.20), a truly explicitscheme was obtained by lumping the mass matrix, which is denoted by ML.The first option, (4.19), consists of solving the pressure–continuity equation


to obtain the nodal values of the pressure at tn+1/2, followed by the solutionof the momentum equations, using the updated pressure pn+1/2, to obtain thevelocities un+1

i . The matrices Kp, Kv are symmetric positive definite and theconjugate gradient method, with, Jacobi preconditioner, is employed to solvethese algebraic systems of equations. Despite the unconditional stability ofthese implicit schemes, the time–step is computed using (4.9) as the accuracydeteriorates rapidly for increasing values of ∆t. The second alternative, (4.20),consists once again of solving implicitly the pressure–continuity equation, butnow followed by the explicit update of the momentum equations.

Slight modifications in equations (4.13) and (4.16) lead to alternative ap-proaches [4] which involve the specification of traction boundary conditions,with the free edge being a special case where zero values are assigned for thetractions.

The introduction of the Poisson–type equation to deal with the pressure,and the automatic presence of the stabilizing parameter in the schemes, elim-inates the zero diagonal term present in standard mixed formulations, whensteady–state conditions are examined. This allows the use of equal–order in-terpolation for all variables, without reference to the Babuska–Brezzi condition[14, 17] to prove convergence of the scheme.

Numerical experiments confirm that the use of equation (4.9) to determine∆t introduces automatically a correct amount of streamline upwinding in theschemes. The stabilizing parameter γ′ introduced in [14] for Stokes’ flow,(Re)E << 1, in the present formulations is related to the time–step by γ′ =∆t/ρ, and is well within the range in which SUPG presents good performance.

4.3 Adaptive Strategies

Transient problems including convection and diffusion effects, such as tran-sient incompressible fluid flows, normally present regions with steep gradientsemerging, moving and disappearing throughout the analysis. An adaptivemesh refinement that follows such features is required in order to have an af-fordable meaningful solution. The strategy adopted for the mesh adaptationuses the computed solution to predict the desired, nearly “optimal”, charac-teristic for a new mesh which is re–generated. This procedure is referred toas remeshing and was proposed by Peraire et al [29]. The key point in sucha procedure is the utilization of a grid generator which encompasses accurategeometry modelling, and the flexibility to generate high quality meshes withsuitable point distribution dictated by the error estimators. The advancingfront technique [28, 29] is adopted in this work to build consistent meshes asan assembly of simple linear triangles. As the analysis algorithms employed al-


low equal–order interpolation, the resultant triangulation is used to discretiseall flow variables.

A rigorous measure of the error, for the coupled set of non–linear partialdifferential equations under consideration, should involve all independent vari-ables. The definition of such norms are very complex and has scarcely beenadvanced beyond a mathematical framework [18, 31, 32]. Such additional com-plexity is avoided by the use of an heuristic adaptive strategy based upon theestimate of the error in the velocity gradients, which at least should give anindication of the trends in the error throughout the computational discretedomain. This provides the information required to generate a new mesh whichwill, in most cases, lead to a solution satisfying the desired quality.

It is important to remark that a criterion based upon the velocity gradientrefers only to the spatial discretisation and is well suited for a mesh adaptivescheme for steady–state solution. When dealing with transient processes, theoverall discrete error in the solution must include the error involved in the timeintegration. Therefore, some form of time–step adaptation is also desirable.This is somewhat embedded in the analysis algorithms since the time–step,equation (4.9), is computed according to the mesh spacing, the physical prop-erties of the flow and the velocity field, in an attempt to optimize the timeintegration process.

4.3.1 Error estimator

Following the idea presented by Zienkiewicz and Zhu [38], and already dis-cussed in chapter 3, the L2–norm of the error in the velocity gradients can beestimated, for a given element E, according to

‖ e⋆E ‖L2=

2∑

i=1

[ ∫

ΩE

(∇u⋆i −∇ui)

T · (∇u⋆i −∇ui) dΩ

]1/2

(4.21)

where ΩE is the subdomain associated with the element E. The discrete L2-norm of the velocity gradient over element E is evaluated as

‖ ∇u⋆E ‖L2=

2∑

i=1

[ ∫

ΩE

∇u⋆i · ∇u⋆

i dΩ]1/2

(4.22)

where ∇ui denotes the discontinuous finite element approximate nodal gradi-ent while ∇u⋆

i is the smoothed continuous nodal gradient, see section 3.2.3.The overall error estimator and the L2–norm of the velocity gradients can becomputed as the summation of the individual element contributions, accordingto


‖ e⋆ ‖2L2

=m∑

E=1

‖ e⋆E ‖2

L2and ‖ ∇u⋆ ‖2

L2=

m∑

E=1

‖ ∇u⋆E ‖2

L2(4.23)

with a normalised error in the solution defined as

η =‖ e⋆ ‖L2

‖ ∇u⋆ ‖L2

(4.24)

4.3.2 Remeshing

It is possible to demonstrate that the requirement of error equidistributionleads to a near “optimal” mesh for elliptic problems [7]. Such a criterion hasalso been applied with success for other classes of problems [28, 36] and isadopted here. Considering a prescribed error tolerance η, this requirementpermits the definition of a target error per element em as

em = η‖ ∇u⋆ ‖L2√

m(4.25)

For elliptic problems, a theoretical convergence–order for uniform meshrefinement can be predicted [36] according to

O(hq) = O(hmin(g,χ)) (4.26)

where g is the order of the shape function adopted in the finite element approx-imation and χ is the strength of the singularity present in the solution. Thedefinition of χ is not a simple task and is problem dependent. Furthermore, itwas found in practice [36] that q = g can be assumed for all elements exceptthose very closed to the singularity, when the mesh with error equidistribu-tion is approached. This allows us to relate the element error for the linearapproximation employed to the current element sizes in the mesh as

‖ e⋆E ‖(l)

L2= ch

(l)E for E = 1, . . . , m (4.27)

where the superscript (l) indicates the current mesh and hE is the minimumheight of the triangular element E. The assumption represented in equation(4.27) was adopted here based on the previous success obtained with such asimplification. On the other hand, it is desirable that the element errors in thenew mesh (l + 1) must be equal to em, which leads to

‖ e⋆E ‖(l+1)

L2= em = ch

(l+1)E for E = 1, . . . , m (4.28)


Equations (4.27), (4.28) can be combined to predict an element size distri-bution for a new mesh such that the pre–computed target element error shouldbe satisfied, i.e.

h(l+1)E = h

(l)E

em

‖ e⋆E ‖(l)

L2

for E = 1, . . . , m (4.29)

This criteria can indicate either local refinement or derefinement and gives adiscontinuous distribution of element sizes throughout the domain. The sameprojection procedure employed to obtain the smooth gradient of velocity ∇u⋆

i

is used to obtain the continuous element sizes necessary as an input to themesh generator. As the error estimator, and hence the target element errorem, is based on the approximate solution, such quantities will only be as goodas the computed solution and there is no guarantee that the specified toleranceη is actually going to be satisfied on the new mesh. Furthermore, the meshgenerator can not always produce the exact element size distribution desired.Thus, a criterion to define the necessity for a mesh redefinition based on theoverall error η has to be checked after each new analysis to verify if furtherremeshing is necessary. In most cases, however, provided that the quality ofthe jump from one mesh to the next is not too large, the strategy describedabove proves successful after the first remeshing for steady–state computations[22, 35].

Treatment of singularities

Some technical issues must be considered when the adaptive remeshing strat-egy is used to deal with problems in which the solution contains singularities.The first point, concerns the use of the asymptotic convergence rate criterionat the element level, represented in equations (4.27), (4.28), which is not realis-tic in the presence of singularities. However, the fact that, when the “optimal”mesh is approached the singularity influence is localized and ceases to domi-nate the overall convergence [37] justifies the generic good performance of themesh regeneration based on (4.27) to (4.29).

Nevertheless, the effects of the singularity will still remain in the vicinity ofthe singularities and the element size distribution indicated by (4.29) may leadto extremely small elements there. On the other hand, elements which are toolarge may be prescribed in regions where the solution is smooth. In practice,it is necessary to limit the computed element sizes between minimum andmaximum acceptable values hmin and hmax, respectively. This is particularlyimportant with regard to the minimum size, as the time–step computed using(4.9) might become too small for an affordable computation. Also note that,essentially for transient problems which contain singularities, the appearance


of local flow features might not be detected by the criterion based on the overallpercentage error [22, 35], as this error depends mainly on the element size inthe neighborhood of the singularities. We follow and adapt the idea proposedby Lyra et al [21], and already discussed in chapter 3, where the mask effectassociated with the singularities is eliminated automatically by purging theircontributions to the computation of the overall error. A two–stage procedureto define the element size distribution can be summarized as:

Stage 1 If the overall normalised error η is larger than the prescribedvalue η then determine the element size distribution for the new mesh usingequation (4.29), but also subject to the cut–off values hmin and hmax as alreadymentioned.

Stage 2 Subdivide the domain Ω according to

Ω• = ∪ΩE |h(l+1)E = hmin

Ω = Ω − Ω•

(4.30)

where h(l+1)E = h

(l)E if in the first stage η was found smaller or equal to η.

The L2–norm of the error (‖ e⋆ ‖L2)Ω in the non–purged region Ω iscomputed as described in equation (4.23), however the summation extend onlyto the m elements inside Ω. The normalised error in this region is definedby

η =(‖ e⋆ ‖L2)Ω

(‖ ∇u⋆ ‖L2)Ω

(4.31)

If the percentage of the error η is larger than the prescribed value η itimplies that the element sizes must be redefined inside the non–purged areausing

h(l+1)E = h

(l)E

e m(

‖ e⋆E ‖(l)

L2

)

Ω

with e

m = η (‖ ∇u⋆ ‖L2)Ω√m

(4.32)

for E = 1, . . . , m. The cut–off values are again applied, i.e. the new sizemust satisfy: hmin ≤ h

(l+1)E ≤ hmax ∀ΩE ∈ Ω and the procedure rendered

safe by not allowing the element sizes recomputed in the non–purged area togrow with respect to the sizes computed in the first stage. This also avoidsupsetting the overall accuracy requirement η = η which is the target at thefirst stage.


4.3.3 Accumulation of interpolation errors

In an adaptive remeshing algorithm it is often necessary to interpolate quan-tities available on a mesh onto a newly generated mesh. During a transientanalysis, a large number of such interpolations may be required, resulting inan accumulation of the interpolation error. To date, this problem has receivedlittle attention in the literature. In reference [6] it was shown that the useof a low order interpolation procedure can badly compromise the overall per-formance of a transient adaptive procedure, regardless of the accuracy of theanalysis algorithm. The whole demonstration is repeated here in detail.

Consider initially a one–dimensional problem. The time derivative of aflow variable at a mesh node may be computed from

∂up

∂t≈ un+1

p − unp

∆t(4.33)

which represents a conventional finite difference approximation. Everything isfine when the same mesh is used at time–levels tn and tn+1. In this case, theaccuracy of the computation depends solely on the accuracy of the algorithmchosen to update the flowfield. On the other hand, we are often faced with theproblem of computing un+1

p on a new mesh while un is not available on thismesh and should be interpolated using the nodal values on the old mesh. Thetime derivative, in practice, is computed according to

∂u′p

∂t≈ un+1

p − unp

∆t(4.34)

where unp denotes an interpolated value.

2un

1un

x=x2x=x1

pun

x=x p

Figure 4.1: Linear interpolation with 1–D linear element.

When linear elements are adopted, an obvious choice is to obtain unp by

using a Lagrangian interpolation employing the linear shape functions and thenodal values, see figure 4.1, as


unp =

(x2 − xp

x2 − x1

)un

1 +(

xp − x1

x2 − x1

)un

2 (4.35)

The error committed in such procedure can be identified with the use ofTaylor series, since

un(xp) = un(x1) + (xp − x1)∂un

∂x

∣∣∣∣x1

+(xp − x1)

2

2

∂2un

∂x2

∣∣∣∣x1

+ h.o.t.

un(xp) = un(x2) + (xp − x2)∂un

∂x

∣∣∣∣x2

+(xp − x2)

2

2

∂2un

∂x2

∣∣∣∣x2

+ h.o.t.

(4.36)

and noting that un(xk) = unk . Equations (4.35)–(4.36) can be combined,

yielding

unp = un

p −(x2 − xp)(xp − x1)

(x2 − x1)

[∂un

∂x

∣∣∣∣x1

− ∂un

∂x

∣∣∣∣x2

+(xp − x1)

2

∂2un

∂x2

∣∣∣∣x1

+(x2 − xp)

2

∂2un

∂x2

∣∣∣∣x2

]+ h.o.t.

(4.37)

Using once again a Taylor series, this time centered at x = xp, we have

∂un

∂x

∣∣∣∣x1

=∂un

∂x

∣∣∣∣xp

+ (x1 − xp)∂2un

∂x2

∣∣∣∣xp

+ h.o.t.

∂un

∂x

∣∣∣∣x2

=∂un

∂x

∣∣∣∣xp

+ (x2 − xp)∂2un

∂x2

∣∣∣∣xp

+ h.o.t.

(4.38)

∂2un

∂x2

∣∣∣∣x1

=∂2un

∂x2

∣∣∣∣x2

=∂2un

∂x2

∣∣∣∣xp

+ h.o.t. (4.39)

Inserting (4.38) and (4.39) into (4.37) results in the relation

unp = un

p +(x2 − xp)(xp − x1)

2

∂2un

∂x2

∣∣∣∣xp

+ h.o.t. (4.40)

Thus from (4.34) and (4.40), we deduce that


un+1p − un

p

∆t︸︷︷︸this is whatis actuallycomputed

=un+1

p − unp

∆t︸︷︷︸this is what

we would liketo compute

− (x2 − xp)(xp − x1)

2∆t

∂2un

∂x2

∣∣∣∣xp︸︷︷︸

this is numerical diffusionassociated with the linear

interpolation

+ h.o.t. (4.41)

From (4.41) it is clear that the linear interpolation unp introduces numer-

ical diffusion, which vanishes if xp coincides with a node in the old mesh.The amount of this diffusion can be analysed for the 1–D convection–diffusionequation

ρCp∂T

∂t+ ρCp|u|

∂T

∂x− K

∂2T

∂x2= 0 (4.42)

where the time derivative, and so all terms of equation (4.41), must be multi-plied by ρCp. This leads to a maximum numerical diffusion equal to

(ρCph

2

8∆t

)∂2T (xp)

∂x2(4.43)

which happens when point P is in the middle of the element x1x2 which is ofsize h. For the pure diffusion limit (Pe → 0), the time–step computed usingequation (4.9), replacing the element reynolds number (Re)E by the elementPeclet number (Pe)E ,[6], is

∆t =ρCph

2

6K(4.44)

and the numerical diffusion (4.43) becomes

(3K

4

)∂2T (xp)

∂x2≡ numerical diffusion for Pe → 0 (4.45)

which is almost equal to (three quarters of) the actual physical diffusion. Forconvection dominated flows (Pe ≫ 1) the time–step , [6], is given by

∆t =Pe − 2

Pe

h

|u| with Pe =ρCp|u|h

K(4.46)

leading to a numerical diffusion, (4.43),

K

8

(Pe2

Pe − 2

)∂2T (xp)

∂x2≡ numerical diffusion for Pe ≫ 1 (4.47)


For instance, if Pe = 10 in equation (4.47) we have 25/16 times the physicaldiffusion and the solution is meaningless. Of course, in the remeshing proce-dure the position of the new nodal points will not very frequently be in theworse possible location, but this parametric study reinforces the importanceof the interpolation stage in the final accuracy of the whole scheme.

In order to avoid an analogous problem in the two–dimensional adaptiveprocedure, a second–order interpolation was developed. This interpolationmakes use not only of the values defined on the triangular element nodes, butalso the smoothed gradients already computed in the error estimator routinesfor the velocity field.

2un

1un

un3

P

Figure 4.2: Interpolation within a linear triangular element.

Suppose u is a generic component of the vector of primitive variables U

and that the cartesian directions are represented by x and y, for notationalconvenience in this section. Let un

p be a linear interpolation using the triangleshape functions Nk and the nodal values un

k indicated in figure 4.2, i.e.

unp =

3∑

k=1

Nk(xp, yp)unk (4.48)

Using a Taylor series in the neighbourhood of node k, we have that


un(xp, yp) = un(xk, yk) + (xp − xk)∂un

∂x

∣∣∣∣k

+ (yp − yk)∂un

∂y

∣∣∣∣k

+(xp − xk)

2

2

∂2un

∂x2

∣∣∣∣k

+(yp − yk)

2

∂2un

∂y2

∣∣∣∣k

+(xp − xk)(yp − yk)∂2un

∂x∂y

∣∣∣∣k

+ h.o.t.

(4.49)

Making use of equations (4.48) and (4.49), we obtain the interpolatedvalue un

p , correct to second–order, after some algebraic manipulation, from therelation

unp = un

p +3∑

k=1

Nk(xp, yp)Rk (4.50)

with

Rk = (xp − xk)∂un

∂x

∣∣∣∣k

+ (yp − yk)∂un

∂y

∣∣∣∣k

+(xp − xk)

2

2

∂2un

∂x2

∣∣∣∣k

+(yp − yk)

2

∂2un

∂y2

∣∣∣∣k

+ (xp − xk)(yp − yk)∂2un

∂x∂y

∣∣∣∣k

(4.51)

where the higher–order terms have been neglected and the second–order deriva-tives in the above equation are computed by differentiation of the smoothedcontinuous gradients already available. If we do not consider the second–orderterms in (4.51), the interpolation is equivalent to an Hermite–type interpo-lation. Note that this second-order interpolation un

p is obtained by adding acorrection term to the first–order linear interpolation un

p . In some cases, itmight be convenient to leave a certain amount of numerical dissipation in thealgorithm by not adding the full correction term. This can be accomplishedby defining a parameter ξ which can be adjusted within the interval [0, 1] andrewriting (4.50) as

unp = up

n + ξ3∑

k=1

Nk(xp, yp)Rk (4.52)

This then represents a linear interpolation when ξ = 0 and a quadratic interpo-lation when ξ = 1. The following procedure is adopted: when an interpolation


is required within an element whose size is hmin, a linear interpolation schemeis adopted choosing ξ = 0 in equation (4.52). There is a good reason to doso, because the hmin value is imposed arbitrarily for computational efficiency,we cannot be sure whether or not a finer mesh is needed in that region. If thegradients there are too steep to be resolved in the distance hmin, a quadraticinterpolation is expected to produce over and under shoots. On the remain-ing parts of the domain, however, the element sizes are computed freely andthere is no reason not to use a more accurate quadratic interpolation. This isaccomplished by setting ξ = 1 in (4.52). It should be observed that the use ofthis strategy might be even more important when dealing with compressibleflow applications. In such cases, the presence of discontinuities in the solu-tion preclude the use of quadratic interpolation in the vicinity of these flowfeatures.

Finally, the conservative nature of the procedures involved in the solutionof compressible flow simulations is of paramount importance, as will be dis-cussed in section 5.3.2. Although, the proposed interpolation procedure cannot be proved to be exactly conservative, the errors involved, when comparedto simple linear interpolation, are certainly of higher order.

4.3.4 The adaptive algorithm for steady and unsteady

flows

The strategies employed for steady–state and transient simulations can besummarized in the following algorithm:

1. Build the initial discrete model;2. Evaluate the current local time–step ∆t = f(hE, (Re)E, u) and ∆tmin;3. Advance the flowfield to the next time–level, (4.19) or (4.20);4. Estimate the overall and non–purged normalised errors;5. If η and η satisfy the required quality then:

5.1 Output results at the current time–level;5.2 If the current time exceeds maximum time required or

steady–state is reached, stop.5.3 Go to step 2;

6. Compute new element sizes distribution, using the two–stageprocedure;

7. Generate a new mesh with the advancing front technique;8. Return to the previous time–level;9. Interpolate variables and boundary conditions onto the new mesh;10. Go to step 2;


The main characteristic of the adaptive remeshing procedure is the com-plete detachment of the stages involved, i.e. generation of the discrete model,analysis algorithm, post–processing error estimation and mesh parameters def-inition are all done independently. The analysis algorithm remains with itsoriginal data structure and once a suitable mesh generator is available theadaptive remeshing procedure can be directly incorporated to many differentfinite element or finite volume programs. Of course, the error estimation stagemust be conveniently developed for each specific application. Some other re-marks concerning practical issues related to this algorithm are pertinent:

Remark 1 - The steady–state strategy is a particular case of the abovealgorithm where the error is checked only after the solution, computed througha pseudo–transient using the elementwise time–step defined according to (4.9),reaches the stationary state on the present mesh. For the transient appli-cations a global time–step ∆tmin, equal to the minimum local time–step,is adopted to integrate the momenta equations. The elliptic nature of thepressure–continuity equation allows the computation of the stabilizing param-eter γ′ = ∆t/ρ elementwise according to the local time–step given by (4.9),even for the unsteady analysis. In fact, this is necessary in order to get a re-alistic pressure distribution, as the use of the smallest local time–step leads toa lack of stability of the pressure solution in regions where the local time–stepis much larger than the global time–step, see numerical example 4.4.1.

Remark 2 - It must be noted that the memory requirement for thesegregated velocity–pressure/momenta algorithms, (4.19) or (4.20), dependson the number of discrete points on the mesh instead of number of degrees offreedom in the problem.

Remark 3 - It should be stressed that the threshold value hmin is de-fined for computational efficiency and limits the overall percentage of errorattainable in the adaptive procedure. Thus the percentage of errors, η and η

might not be achieved unless hmin is sufficiently small.

Remark 4 - In a transient analysis, it is very important to reset thetime after each remeshing to guarantee the solution quality. Another aspectof the present strategy is that a new remeshing is only allowed if the currenttime is bigger to the time of the last remeshing.

Remark 5 - In order to interpolate the pressure field also with a second–order procedure, a continuous pressure gradient must be computed, using thesame procedure adopted for the velocity components. Nevertheless, for thecases analysed, this was found to be unimportant, and only the velocity iscomputed with higher–order interpolation. This represents just a small in-crease in computation cost since the continuous velocity gradients are alreadycomputed in the error analysis routine and kept in memory.


4.4 Numerical Examples

In this section, the effectiveness of the proposed approach is demonstrated fortransient and steady–state simulations. The applications were chosen to ex-ploit the full capacities of the Petrov–Galerkin discretisation, segregated solu-tion procedure and the adaptive remeshing strategy. The results presented herewere obtained using the pressure–continuity/Taylor–Galerkin scheme given in(4.20), which integrates the momentum equations explicitly while the pressure–continuity equation is solved through the conjugate–gradient method with aJacobi preconditioner. Some of the technical difficulties, for instance pres-sure oscillations, presence of singularities and interpolation error, faced duringthe development of the final procedure are also addressed and the strategiesadopted to overcome those problems are shown to play a fundamental role inthe success of the examples analysed.

4.4.1 Steady–state leaky–lid driven cavity flow

The lid–driven cavity flow problem represents a classical test case [2] for incom-pressible Navier–Stokes simulation with or without heat transfer. The problemof interest here is the unit square cavity in the absence of heat transfer anda Reynolds number equal to 100, with the problem definition given in figure4.3(a). The velocity components are initially prescribed to be zero, exceptat the top boundary where the lid moves with V = 1.0 so that the horizon-tal velocity component is set to a unit value. As mentioned in section 4.2, areference pressure has to be specified in order to avoid indeterminacy of thepressure field. A zero pressure value has been imposed at the mid–point of thebottom of the cavity. The singularities present at the top corners [30], wherethe horizontal velocity component u1 is not uniquely defined, are regularizedby considering a leak close to the cavity lid [35]. This was done by assuminga linear variation of u1 in a fixed region in the vicinity of the top corners.

The initial mesh used in the analysis is shown in figure 4.3(b), whichrepresents a coarse, nearly homogeneous mesh containing 220 elements and133 nodes. No attempt has been made to design this mesh in a clever wayso that the importance of the adaptive procedure can be emphasised. Theoverall percentage error in the gradient of velocity representation η, for thesteady–state solution, obtained on this mesh is 40.3%. The velocity vectorsand pressure contours obtained with the initial mesh can be seen in figure 4.4.

Target errors η = 25% and η = 20% were prescribed, which led to therefined mesh shown in figure 4.5 after applying the two–stage procedure tocompute the spatial distribution of spacing in the initial mesh.

The refined mesh contains 632 elements and 358 nodes. The overall per-centage of error and the percentage of error in the non–purged subdomain in


(a) (b)

1.0

1.0

0.05

P=0

V = 1

Figure 4.3: The cavity flow problem: (a) problem definition; (b) initial mesh.

(a) (b)

Figure 4.4: The cavity flow problem results for the initial mesh. (a) Velocityfield; (b) pressure field.

the new analysis are 15.6% and 14.4%, respectively. These conservative val-ues are mainly a result of the simplified assumption inherent in (4.29), factorsrelated to the mesh generation algorithm and the relatively big jump betweenthe error in the original mesh and the target error. However, the desired tar-gets were achieved after a single refinement in all steady–state applicationswhich were performed. This might not be true if the refinement had not beendone in a safe way.

When a transient analysis of the same problem was attempted it wasfound that the use of the smallest local time–step for the whole domain ledto a lack of stability of the pressure solution, as already mentioned in section


Figure 4.5: Refined mesh for the lid–driven cavity flow problem.

(a) (b)

Figure 4.6: The cavity flow problem results computing γ′ globally for thepressure–continuity equation. (a) Velocity field; (b) pressure field.

4.3.4. For the initial mesh, this problem is not observed as the mesh is almostuniform. This behavior can be seen in figure 4.6, where although the velocityfield looks “normal”, the pressure field shows unrealistic oscillations.

The solution to the above problem is, however, rather simple and con-sists in computing the parameter γ′ = ∆t/ρ with the local time–step in thepressure–continuity equation, as already mentioned in section 4.3.4. The globaltime–step is still used to integrate the momentum equations and the resultsnow are illustrated in figure 4.7. Interestingly enough, the velocity field infigure 4.7(a) is very similar to the one in figure 4.6(a), but note now the stablepressure field shown in figure 4.7(b). It is also immediately apparent that theresults on the adapted mesh are an improvement when compared with thatobtained with the initial mesh.


(a) (b)

Figure 4.7: The cavity flow problem results computing γ′ locally for thepressure–continuity equation. (a) Velocity field; (b) pressure field.

4.4.2 Simulation of Von Karman vortex street behind a

circular cylinder

The problem of a circular cylinder in a crossflow is analysed as a second appli-cation. For a Reynolds number Re based on the freestream velocity u∞ andon the cylinder diameter, experiments show that a steady–state solution withsymmetric vortices appearing behind the cylinder occurs up to Re ≈ 40, [26].However, for higher Reynolds numbers, the flow shows a periodic sheddingof vortices, forming what is called a Von Karman vortex street. Even higherReynolds numbers will lead to a turbulent wake, but this is beyond the scopeof laminar flow simulation analysed here.

The numerical simulation of this complex flow is a challenging test for theadaptive solution procedure. First, all terms in the momentum equations playan important role in establishing the periodic character of the flow [1] andthis demands an accurate discretisation. Secondly, the presence of a strongsingularity at the front of the cylinder and the need to use a large number ofmeshes serves to test the performance of the remeshing procedure with regardto the difficulties outlined in previous sections. To pass the test the algorithmhas to be able to automatically detect the downstream formation of the vortexstreet, regardless of the presence of the strong singularity at the stagnationpoint. It also has to lead to meaningful results, despite the large number ofinter–mesh interpolations involved.

Figure 4.8 shows the domain specification and boundary conditions con-sidered. This problem represents an external flow, and providing the inletis taken away from the obstacle, the freestream velocity can be imposed asa boundary condition there. A constant pressure, p = 0, and free traction


16.0

4.0

4.0

8.0

2u 0=

1u 0=2u 0=

2u 0=

2u 0=1u 1=

0=t 1

0=t 1

Xq P 0=

0=t 1

0=2t

Figure 4.8: Geometry definition and boundary conditions for the cylinder ina crossflow problem.

boundary conditions are prescribed at the outflow. The side external bound-aries are positioned at a distance considered reasonable to avoid tunnel effectsand the boundary conditions are as shown in figure 4.8. The initial conditionconsists of an uniform horizontal velocity field with u1 = 1 everywhere, exceptat the cylinder surface, where non–slip conditions have been assumed. Theabove initial condition violates the requirement that the fluid is always andeverywhere incompressible, leading to an ill-posed problem [10]. However, thisdoes not present any problems for the algorithm as the first step produces asmooth divergence free velocity field, which becomes the effective initial con-dition.

Figure 4.9: Fixed fine mesh for the cylinder in a crossflow problem.

First, some numerical results for Re = 100, obtained on a fixed mesh,with no adaptation, are presented. The mesh, containing of 3317 nodes and


6468 elements, is shown in figure 4.9. Note that this fixed mesh has been builtusing previous knowledge about the formation of the vortex street behind thecylinder, where the mesh is much more refined. The same computation on afixed homogeneously fine mesh would have been extremely costly and, for thatreason, has not been considered.

Figure 4.10(a) shows the variation of the velocity component u2 at pointq on the outlet boundary, see figure 4.8. Note that after the initial transient,the velocity component u2 exhibits a periodic behavior. The frequency ofthe oscillation is in accordance with available experimental [26] and numericaldata [1]. The evolution of the estimated overall and purged normalised errorsduring the analysis can be seen in figure 4.10(b). Observe that this error ishigh for a short period at the beginning of the analysis. During this phase,the fixed mesh is not capable of accurately representing the singularity at thefront of the cylinder and the formation of the boundary layers, despite the useof a finer grid in this region. The estimated error reaches a minimum prior tothe onset of the periodic vortex shedding, but rises a little again, showing aperiodic behavior when this is finally established.

(a) (b)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80

t

2u

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80

GlobalLocal

t

η

Figure 4.10: The cylinder in crossflow problem on a fixed mesh. (a) Verticalvelocity at point q; (b) history of the estimated errors η and η.

The same problem was analysed with the adaptive remeshing, assumingη = 22% and η = 15%, but using simple linear interpolation to interchangedata within meshes. The vortex street formation is properly identified, despitethe presence of the singularity at the front of the cylinder. A total number of114 meshes was employed while advancing the solution from t = 0 to t = 80.About 8 meshes are constructed during each oscillation period, when periodicflow was established. Both velocity and pressure fields, and meshes obtained,showed qualitatively reasonable results. However, when the periodic behaviorof the velocity component u2 at the point q is plotted, a non–constant and


generally damped amplitude is observed as indicated in figure 4.11(a). This canbe explained by the accumulation of interpolation errors discussed in section4.3.3. The leading error associated with the linear interpolation plays therole of a spurious anisotropic numerical viscosity. This explain the generallydamped amplitude results obtained. However, as this spurious term appearsat random at each interpolation, it also explains why the amplitude of u2 isnot damped smoothly.

(a) (b)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80

t

u 2

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80

t

2u

Figure 4.11: Vertical velocity at point q with adaptive remeshing for the cylin-der in crossflow problem: (a) linear inter–meshes interpolation; (b) selectivehigher–order inter–mesh interpolation.

Finally, the results of an adaptive solution procedure, where a selectiveinterpolation, (4.52), is used to relate the variables on different meshes, arepresented. Figure 4.11(b) depicts the periodic behavior of the velocity compo-nent u2, in which the problems regarding the damping and non–uniformity ofamplitude have been corrected to a large extent.

In this second adaptive simulation, a total number of 215 meshes wasused to update the solution from t = 0 to t = 80. About 18 meshes wereautomatically generated during each oscillation period. The meshes at thestart of the analysis typically contained 1000 nodes and 2000 elements. Thisnumber doubles, but remains nearly constant, when the periodic vortex streetis established. The oscillation period obtained numerically is approximatelyequal to six, which is in agreement with existing experimental and numericaldata [1, 26]. Finally, in figure 4.12, the expected anti–symmetric character ofthe velocity field obtained at two instants, roughly separately apart by halfperiod, can be observed.

The results at times t = 5, 15, 55 are presented in figures 4.13 to 4.15.Figure 4.13 shows the results at time t = 5. It can be seen that there is theformation of a thin boundary layer close to the cylinder with a separation at


(a)

(b)

Figure 4.12: Cylinder in a crossflow problem. Velocity vectors at instantsdiffering in time of half period.

the back to form a pair of recirculating symmetric eddies. The refinementconcentrates on the vicinity of the cylinder, with an almost uniform coarsemesh away from it as the solution remains smooth there.


(a)

(b)

(c)

(d)

Figure 4.13: Adaptive results for the cylinder in a crossflow problem at timet = 5. (a) Mesh; (b) velocity vectors; (c) streamlines and (d) pressure contours.


(a)

(b)

(c)

(d)

Figure 4.14: Adaptive results for the cylinder in a crossflow problem at time t =15. (a) Mesh; (b) velocity vectors; (c) streamlines and (d) pressure contours.


(a)

(b)

(c)

(d)

Figure 4.15: Adaptive results for the cylinder in a crossflow problem at time t =55. (a) Mesh; (b) velocity vectors; (c) streamlines and (d) pressure contours.


In figure 4.14, the solution is shown for time t = 15. At this time–level,the initially symmetric eddies become elongated in the flow direction and areno longer in balance. This is observed by looking at the unsymmetric natureof the plotted results. At time t = 55, figure 4.15, the Von Karman vortexstreet is already developed and the periodic flow is fully established. The twoparallel row of staggered vortices can best be appreciated from the pressurecontours.


There is a general feeling among CFD practitioners that a detailed, and yetaffordable, computation of complex industrial fluid flows problems will only bepossible by the combination of an accurate solution algorithm and an efficientadaptive technique. A comprehensive strategy for the adaptive solution ofthe incompressible Navier–Stokes equations has been presented. It includes aPetrov–Galerkin method which automatically introduces streamline upwindingand the stabilizing terms required to avoid Babuska–Brezzi restrictions. Thisallows for the use of simple linear triangular elements to approximate bothvelocity and pressure. Therefore, the element matrices can be computed ana-lytically, which leads to improved computational efficiency, which is importantespecially if a 3–D extension is considered.

In general, the matrix corresponding to the mixed formulations is non–symmetric due to the convection terms in the momentum equations and non–positive–definite owing to the uncoupled nature of the continuity equation [15].The solution of this class of equations by direct methods is prohibitive due tomemory requirements and the solution by available iterative methods is notsufficiently robust for many practical applications. The symmetric positivedefinite nature of the matrix involved in the presented formulation representsanother attractive characteristic of the least–squares formulation, as the robustand efficient preconditioned conjugate gradient method can be used. Further-more, the pressure–continuity/least–squares solution algorithm which solvesboth momentum and pressure–continuity equations implicitly constitutes apromising alternative for vector and parallel computation. This results fromthe fact that the CPU time required for a computation is dominated by theroutine utilized for the solution of the algebraic system of equations and effi-cient implementations of the conjugate gradient method with a vector [3, 13]or parallel [34] machines have already been demonstrated.

The accuracy of the results was found to be extremely dependent on spe-cific features inherent in the adaptive remeshing procedure, independent of theorder of accuracy advocated for the analysis algorithm itself. The concept ofpurged error was employed in order to be able to detect the flow details, despite


the masking effect associated with strong singularities. This has worked wellfor the example of the cylinder in a crossflow, with all the remeshing requiredto capture the formation of the vortex street being produced automatically,regardless of the strong fixed singularity that exists in this problem. Anotherimportant aspect concerns the accumulation of interpolation errors inherentin any adaptive remeshing strategy. The spurious dissipation caused by thestandard linear interpolation has been shown both analytically and numeri-cally. A simple and inexpensive quadratic interpolation has been developed toovercome this difficulty and this reduces the interpolation effects to an ordercompatible to that of the analysis algorithm.

The refinement and coarsening are directly embedded in the remeshingscheme, with the number of elements not increasing so rapidly as with a puremesh enrichment procedure, and the control on the element quality makes thisscheme very powerful for transient analysis. The remeshing procedure alsoallows readily for the capture of 1–D features in an efficient way by defining amesh stretching parameter [29]. This makes the approach very attractive whenattempting to simulate compressible flows. Another improvement possible fortransient applications, relates to the automatic definition of regions, througherror criteria, where the mesh must be redefined and the remainder of thedomain is left without changing the grid. This “refinement” of the remeshingstrategy has been successfully applied for transient flows involving movingbodies [28] and will certainly lead to CPU savings in other situations.

The stability of the adopted formulation, regardless of the interpolationadopted for pressure and velocity, opens the possibility of the use of adaptivehierarchic p or h–p versions of the finite element method [15, 25]. Anotherissue worthy to be considered, is connected with the enhancement of the effi-ciency of the described procedures. In this respect, the use of time partition-ing techniques [11, 20], i.e. adaptive time integration, for transient analysiswith time–steps of different magnitude or the adoption of different integra-tion techniques [33] within automatically defined regions of the domain, arevery promising. Finally, the procedures introduced are extendible both to theanalysis of 3–D problems and the modeling of more complex incompressibleflows. The simulation of natural and mixed convection can be easily achievedby the inclusion of the energy equation plus buoyancy terms [37]. The analysisof turbulent flows can also be accomplished within the framework developedhere.

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Chapter 5

1-D Hyperbolic Equations &Upwind Methods

5.1 Introduction

Many problems in engineering and science are governed by conservation lawsexpressed in terms of hyperbolic partial differential equations, such as thesystem of Euler equations which are of importance in computational fluid dy-namics. This particular class of partial differential equations presents someproperties which must be well understood before attempting to develop nu-merical methods for their solution. Some elements of the theory of HyperbolicPDE are, in some way, incorporated by most successful numerical schemesavailable, designed to solve such problems. In particular, the physical propa-gation of perturbations along the characteristics, which is typical of hyperbolicequations, plays a major role in the family of numerical schemes referred to asupwind schemes. The robustness of upstream discretisation, the availabilityof a physical interpretation, and the possibility of achieving high resolution ofstationary discontinuities, means that upwind schemes have achieved a highpopularity among computational fluid dynamics practitioners. The first partof this chapter covers a few elementary concepts of the theory of HyperbolicPDE, and an alternative and more detailed presentation of the theme can befound elsewhere [20, 22, 25, 40, 51, 61]. The remainder of this chapter con-cerns the description of a variety of classical and recent developed upwindapproaches and the study of their properties and performance when solvingproblems governed by the 1-D Euler system of equations.

Hyperbolic PDE & Upwind Schemes 121

5.2 Theoretical Background on Hyperbolic PDE

5.2.1 Characteristic concept

Linear equation

A subset of the Initial Boundary Value Problem, in which no limit in spatialdomain is considered, is referred to as a pure initial value problem (IVP) orCauchy problem [19]. Although it has no real practical application, since onlybounded domains occur in practice, it has certain theoretical importance andwill be used here to introduce some basic concepts. The simplest hyperbolicIVP consists of the linear wave, or advection, equation described in chapter 2,in one space dimension, and can be stated as: Find a solution for

∂u

∂t+ a

∂u

∂x= 0 for −∞ ≤ x ≤ ∞ and t ≥ 0

u(x, 0) = u0(x) for −∞ ≤ x ≤ ∞ at t = 0(5.1)

where a represents the convective velocity.

The rate of change of u(x(t), t) along an arbitrary line Γ in the x− t planeis expressed by

∂u

∂t+

dx

dt

∂u

∂x=

du

dt(5.2)

using equation (5.1) in the above expression, it is easy to check that

du

dt= 0 for all lines Γ such that

dx

dt= a (5.3)

i.e. the solution u(x, t) is constant along the lines

x0 = x − at (5.4)

Such lines are called the characteristics of equation (5.1). Equation (5.3) im-plies that, as time evolves, the initial data is simply propagated unchangedwith velocity a, and the solution at any point (x, t) is simply

u(x, t) = u(x0, 0) = u0(x0) (5.5)

The solution u(x, t) at any point (x, t) depends only on the initial data u0 ata single point x0 = x− at which lies on the characteristic that passes through(x, t). If the initial data should change at any point other than x0, the solution


at point (x, t) will remain the same. This fact means that the domain ofdependence of the solution at the point (x, t) is the point x0, and the range ofinfluence of the the initial data u0 at a single point x0 is limited to the pointsthat lie on the characteristic that passes through x0. For instance, if the initialdata has a singularity (e.q. discontinuity) at a point x0 it will propagate alongthe characteristic curve that passes through x0, but the solution will remainsmooth along characteristics, emanating from smooth portions of the initialdata.

u(x,t)

u(x,0)

t

x0 L

Figure 5.1: Characteristics for the advection equation.

Suppose the problem described in equation (5.1) has a > 0 and a finitedomain Ω = x|0 < x < L. From the previous observations and the analysisof the characteristic curves presented in figure 5.1, it becomes obvious thatthe solution everywhere in the limited x− t plane is determined by the initialcondition and boundary condition at x = 0, where incoming characteristics arepresent. No boundary condition can be imposed at x = L where the character-istics are leaving the domain. This feature forms the basis of the characteristicboundary method, where only a condition on the variable transported from theboundary towards the interior of the domain is physically imposed. However,information on the outgoing boundary is also necessary for the numerical sim-ulation and this information has to be consistent with the physical behavior ofthe flow, as well as compatible with the discretised equation. This can causeadditional difficulties when modelling such numerical boundary conditions [20].

Non–linear equation

In order to study some phenomena which are genuinely non–linear, such asshock formation, let us consider the initial value problem of finding a solution


of

∂u

∂t+ u

∂u

∂x= 0 for −∞ ≤ x ≤ ∞ and t ≥ 0

u(x, 0) = u0(x) for −∞ ≤ x ≤ ∞ at t = 0(5.6)

in which the inviscid Burgers equation is written in a quasi–linear form. Notethat equation (5.6) looks like an advection equation, but with the advectionvelocity equal to the transported quantity u. It can be shown, as for the linearcase, that the solution u(x, t) is constant along the characteristics

x0 = x − u(x0)t (5.7)

which are again straight lines but with different slopes u, and determined bythe initial data from equation (5.5), provided the characteristic do not crosseach other.

u(x,0)

t

x0 L

dxdt

= CL Sdxdt

= dxdt

= CR

shock

> >

u(x,t)

ts

Figure 5.2: Shock formation for inviscid Burgers equation.

Because of the dependence of the solution upon the propagation speed,the initial data shape changes and a discontinuous configuration can evolve tosmooth solution or a discontinuity can be generated from initial smooth data.See figure 5.2 where the initial data, the characteristics and the solution attime t ≥ ts are presented. In this figure it should be noted that certain charac-teristics intersect at time t = ts, as a result of the fact that the characteristic


speed on the left CL is bigger than that on the right CR. This means that amultiple–valued solution results, but in most physical situations this does notmake sense and, in the idealized complete absence of diffusion, a discontinuityis formed at time ts given by

ts =−1

max

[du0(x0)

dx0

] (5.8)

whenever the gradient of u0(x0) < 0. By analogy to gas dynamics, this line,across which the solution changes discontinuously, is called here a shock. Be-yond ts there is no classical solution for the partial differential equation givenin (5.6). The formation of discontinuity such as that in figure 5.2, whichpropagates at speed S, represents a common feature of first–order hyperbolicconservation laws present in fluid mechanics.

5.2.2 Weak form of a conservation law

In order to demonstrate the essential importance of the weak formulation de-scribed in section 2.3.3, when a first–order hyperbolic problem is considered,assume the problem given by equation (5.6) by rewriting in conservative formas: find a solution of

∂u

∂t+

∂F (u)

∂x= 0 for −∞ ≤ x ≤ ∞ and t ≥ 0

u(x, 0) = u0(x) for −∞ ≤ x ≤ ∞ at t = 0(5.9)

Let ωl be any continuous function such that ωl → 0 as |x| → ∞ and at time–level t = T as T → ∞. The weighted residual approximate form (2.13) for theabove problem can then be stated as: find u such that

∫ T

0

∫ ∞

−∞

(∂u

∂t+

∂F (u)

∂x

)ωl dx dt = 0 (5.10)

The weak formulation can be obtained by using the divergence theoremin equation (5.10) and applying the fact that ωl vanishes at the infinity. Theresulting formulation can be described as: find u in TTT such that

∫ T

0

∫ ∞

−∞

(u∂ωl

∂t+ F (u)

∂ωl

∂x

)dx dt +

∫ ∞

−∞u0(x)ωl(x, 0) dx = 0 (5.11)


Since no derivative appears on the approximate solution u and on the fluxF (u) the solution of (5.11) does not need to be smooth, allowing discontinuities.Equation (5.11) represents the weak form of the problem (5.9) and if its solutionu is smooth all over the domain the previous steps can be reversed and it canbe shown that u also satisfies (5.9), i.e. u is a strong solution.

The weak form given in (5.11) is equivalent [18] to requiring that theintegration of the governing equation (5.9) over all rectangles (A, B) × (t1, t2)holds, i.e.

∫ B

Au(x, t2) dx−

∫ B

Au(x, t1) dx+

∫ t2

t1F (u(B, t) dt−

∫ t2

t1F (u(A, t) dt = 0 (5.12)

This represents a simplified weak form obtained using Green’s theorem andthe fact that the domain is a rectangle.

5.2.3 Rankine–Hugoniot relations

Consider that the weak solution u of equation (5.9) is discontinuous across aline ΓS and smooth in the subdomains ΩL, ΩR, whose boundaries are ΓL andΓR respectively (see figure 5.3). Reversing the steps used to obtain equation(5.11), for each subdomain ΩL, ΩR, it reduces to the alternative form

Ω L

Ω

t

xA B

T

88-

ΓR

~ΓL

~

~

ΓS~

Ω R~

Figure 5.3: Sketch of discontinuous weak solution.

∫

ΓL

ωl[udx − Fdt]L +∫

ΓR

ωl[udx − Fdt]R +∫

Ωu0(x)ωl(x, 0) dx = 0 (5.13)


in which the fact that the solution u is smooth inside the subdomains satisfying(5.9) was used and Green’s theorem was employed. The integrals in equation(5.13) vanish when |x|,T→ ∞ as ωl vanishes there, leading to

−∫

Ωωl[udx − Fdt]L +

∫

ΓS

ωl[udx − Fdt]L −∫

ΓS

ωl[udx − Fdt]R

+∫

Ωu0(x)ωl(x, 0) dx = 0

(5.14)

where the negative signs appear as a result of the orientations when computingthe integrals over the boundaries. The first term cancels the fourth term, asu = u0(x) and dt = 0 in Ω at time t = 0, resulting in

∫

ΓS

ωl [uL − uR] dx − [F (uL) − F (uR)] dt = 0 (5.15)

As ωl is an arbitrary function, it can be deduced that

S =dx

dt=

[F (uL) − F (uR)]

[uL − uR]on ΓS (5.16)

This represents the local conservation property over a discontinuity and iscalled Rankine–Hugoniot condition, with S being the speed at which the dis-continuity propagates. For the linear advection equation (5.1) F = au leadingto S = dx/dt = a which coincides with the characteristics curves (5.3) and adiscontinuity can propagates only along the characteristic in this case.

5.2.4 Entropy condition

The solutions of the weak form of the conservation laws (5.11) are unfortu-nately not unique, as can be seen in figure 5.4 where, for the same initial data,two solutions, from infinitely possible weak solutions, are presented. In thefirst case, in figure 5.4(a), the characteristics go out from the shock and apoint on the shock can not be traced back to the initial data. Comparing thecharacteristics of the “compression” shock in figures 5.2 and the “rarefaction”shock in figure 5.4(a) it can be observed that in the first type of discontinuitythe speed of the characteristics satisfy

CL > S > CR (5.17)

which represents Lax’s entropy condition for a genuinely non–linear field [25,51], but in the second type of discontinuity this is not true. The adjectives


“compression” and “rarefaction” used here come from analogy to gas dynamicsequations, [20].

A more general form for the entropy condition introduced by Oleinik [33]can be stated as

F (u) − F (uL)

u − uL≥ S ≥ F (u) − F (uR)

u − uR(5.18)

These entropy conditions correspond, physically, to guaranteeing an entropyincrease through the discontinuity, i.e. satisfaction of the second principle ofthermodynamics [20].

The second solution presented in figure 5.4(b) represents the physicallycorrect solution and corresponds to the solution of the viscous Burgers equation(2.3) in the limit of vanishing the diffusive term

find u satisfying:

limd→0

[∂u

∂t+ u

∂u

∂x= d

∂2u

∂x2

](5.19)

The propagation of an initial expansion shock is not stable since the additionof a small amount of viscosity, or smearing out the initial profile a little, willlead to the expansion fan solution of figure 5.4(b). This result has importancewhen attempting to implement numerically the entropy condition for inviscidflow computations, as it implies that an appropriate amount of artificial viscos-ity must be added to the discretised equations in order to avoid non–physicaldiscontinuities [20]. A theorem linking the entropy condition to the numericalsolutions due to Le Roux [24] can be stated as “If a numerical scheme is con-vergent and satisfies the entropy condition, then it converges to the physicallyconsistent solution”. Thus the entropy condition is a necessary condition tohave meaningful solution.

5.2.5 System of equations

To extend some of the theory presented for scalar equations into hyperbolicsystem of equations, consider the 1–D initial value problem of finding a solutionto

∂U

∂t+

∂F (U)

∂x= 0 for −∞ ≤ x ≤ ∞ and t ≥ 0

U(x, 0) = U 0(x) for −∞ ≤ x ≤ ∞ at t = 0

(5.20)


(a) (b)

u(x,0)

t

x0 L

u L

u R

u(x,t)

dxdt

=dxdt

=CL CR<< dxdt

= S

u(x,0)

t

x0 L

u L

u R

u(x,t)

dxdt

=dxdt

=CL CR

<< dxdt

Figure 5.4: Propagation of an initial expansion discontinuity for inviscid Burg-ers equation: (a) Entropy–violating shock; (b) entropy–preserving expansionfan.

This can be regarded as a problem governed by a system of m coupled conser-vation laws. The system of equations in (5.20) can be alternatively written ina quasi–linear form as

∂U

∂t+ A

∂U

∂x= 0 (5.21)

where A is the Jacobian matrix, ∂F /∂U. The hyperbolicity assumption meansthat all m eigenvalues of the matrix A are real and that a complete set of righteigenvectors rk exist such that the transformation


A = RΛ R−1 (5.22)

is possible, where R denotes the matrix whose columns are the right eigenvec-tors rk of A and Λ = diag(λ1, . . . , λm) is the diagonal matrix which containsthe eigenvalues of A. Inserting equation (5.22) into the system (5.21) andmultiplying by R−1 gives

R−1 ∂U

∂t+ Λ R−1∂U

∂x= 0 (5.23)

Introducing a new set of variables δW, called the characteristic variables, de-fined by the relation

δW = R−1δU or δU =m∑

k=1

rkδWk (5.24)

where δ represents an arbitrary variation, it is evident that, for the problemswhere the previous equation can be integrated, the variables W can be definedallowing us to write the system (5.23) as

∂W

∂t+ Λ

∂W

∂x= 0 (5.25)

This represents a system of m scalar equations and the components of thevector W, also called Riemann variables or invariants, propagate along thecorresponding characteristics with the speed λk. For a linear system of equa-tions, the components of that matrix A are constant and the eigenvectors rk areindependent of U, with the variables W always being defined. The resultantsystem (5.25) is decoupled into m independent linear scalar advection equa-tions and the solution can be obtain using (5.5) for each characteristic variableWk(k = 1, m) along the characteristic curves dx/dt = λk. The solution tothe original system is recovered using the relation between the conservativevariables and the characteristic variables

U = RW =m∑

k=1

rkWk (5.26)

For a non–linear system of equations, where the matrix A = A(U), if the sys-tem is hyperbolic and the variables W are defined, it is still possible to writethe system in the form (5.25) but λk will depend on the solution of the problemU and the system will no longer be decoupled, therefore it will not be pos-sible to solve the problem by first determining the characteristics, solving anindependent set of ordinary differential equations along the characteristics and


later recovering the conservative variables. It should be mentioned that, fornon–linear system of equations, it is not always possible to find the vector W

when the number of dependent variables is more than two and it is not possibleif more than two differentials appear in the linear combination (5.24), which isalways the case for unsteady multidimensional Euler equations [20, 62]. Nev-ertheless, (5.23) and (5.24) still hold true and, if a linearization of the problemabout a constant state U is adopted, a local system of characteristics W canbe defined and a local frozen constant coefficient system (5.25) is obtainedwith Λ(U). In this case, the coupling between the characteristic variables willappear only through the eigenvectors matrix R in relation (5.26). Small dis-turbances can be considered to propagate, approximately, along characteristiccurves of the form dx/dt = λk(U) and this procedure represents the basis forthe local–characteristic approach to be discussed later in this chapter.

5.2.6 The 1–D Euler equations

Consider the system of compressible Euler equations described in section 2.5in a one–dimensional space

∂U

∂t+

∂F1

∂x1

= 0 (5.27)

which is a system of non–linear hyperbolic equations. Dropping the subscriptand superscript 1, the vector of conservative variables and the flux vectors(2.42) reduce to

U =

ρρuρε

F =

ρuρu2 + p

(ρε + p)u

(5.28)

with the jacobian matrix A = ∂F /∂U given by

A =

0 1 0(γ − 3)u2/2 (3 − γ)u (γ − 1)

(γ − 1)u3 − γuε γε − 3(γ − 1)u2/2 γu

(5.29)

The eigenvalues of A are

λ1(U) = u, λ2(U) = u + c, λ3(U) = u − c (5.30)


where c denotes the speed of sound, computed as described in equation (2.51),for a perfect gas.

The flux vector F (U) is an homogeneous function of degree one of theconservative variables vector U [53], i.e. for an arbitrary constant α

F (αU) = αF (U) (5.31)

whenever the equation of state of the fluid satisfies the relation

p = ρf(e) (5.32)

which is the case for a perfect gas (2.48) in gas dynamics. This remarkableproperty implies that Euler’s identity holds, i.e.

F (U) =∂F

∂UU = AU (5.33)

The flux vectors Fj

in the multidimensional inviscid equations of gasdynamics also have the homogeneous property. The equality in equation (5.33)implies that the flux vector F can be split into subvectors associated with a setof eigenvalues and represents the basis for the flux vector splitting approachwhich will be discussed in section 5.4.2.

Characteristic waves

The characteristic variables are only integrable for isentropic or isothermalflows when the 1–D Euler equations are considered. However, the incrementalform (5.24) is always possible assuming a small space–time interval, and thecharacteristic variables δWk propagate along the corresponding characteristics

C1 :dx

dt= u

C2 :dx

dt= u + c

C3 :dx

dt= u − c

(5.34)

Difficulties arise in the boundary condition specification for the Eulerequations because, depending on the flow local regime, it contains eigenvaluesof both signs at the boundaries, implying that waves are propagating into andout of the domain. For a local supersonic flow at the boundaries, u > c, and allinformation is carried in the direction of the flow, see figure 5.5(a). Therefore a


disturbance can not affect upstream and physically three boundary conditionsmust be imposed at an inlet and zero at an outlet. For a local subsonicflow, u < c, the wave corresponding to the characteristic C3 propagates inthe upstream direction and only two physical boundary conditions must beimposed at the inlet and one at the outlet, 5.5(b).

(a) (b)

C1

C1

C2

C2

C3

C3

X

t

PI

PO

X I XO

Ω~

C1

C1

C2

C2

C3 C3

X

t

PI

PO

X I XO

Ω~

Figure 5.5: Propagation of characteristics through the boundaries for one–dimensional inviscid flows. (a) Locally supersonic regime and (b) locally sub-sonic regime.

The non–linear nature of Euler equations allows formation and propaga-tion of discontinuities in the solution. If a frame of reference which is movingwith the discontinuity is considered, (S = 0), the Rankine–Hugoniot relationssubstituting the vectors (5.28) into equation (5.16) lead to

(ρu)L = (ρu)R

(ρu2 + p)L = (ρu2 + p)R

(ρε + p)uL = (ρε + p)uR

(5.35)

where the subscripts L and R represents the left and right states separated bythe discontinuity.

The system of Euler equations has the property that in the characteristicfield C1 the eigenvalue λ1(U) is constant along integral curves of the field [25,51], i.e. the gradient of the eigenvalue λ1(U) is orthogonal to its eigenvectors

∇λ1(U) · r1 = 0 ∀ U (5.36)

where ∇ represents the gradient with respect to the components of U. Thisfield is linearly degenerate and a discontinuity in this field is referred to as acontact discontinuity. In such situations, the discontinuity moves at the samevelocity as the fluid on each side λ1(UL) = S = λ1(UR) and the equality signal


must be added to the entropy condition (5.17), such as in (5.18), to encompassthe contact discontinuities. This equality means that the characteristics areparallel to the propagating discontinuity on both sides, just as in the propa-gation of a discontinuity described for the linear scalar equation in figure 5.1.The Rankine–Hugoniot relations (5.35), considering the moving frame, requirethat

ρL 6= ρR

uL = uR = 0

pR = pL

(5.37)

and if two gases are initially in contact with each other satisfying these condi-tions, they will propagate in contact but without mixing. The specific energyis also discontinuous and also the temperature, which make possible differ-ent densities at the same pressure. When isothermal or isentropic flow isconsidered, the Euler equations reduce to two equations which are genuinelynon–linear and no contact discontinuity is possible.

The other possible discontinuity occurs when the velocities are differentat each side leading to discontinuity in all quantities

ρL 6= ρR

uL 6= uR

pR 6= pL

(5.38)

This represents a shock wave and is connected to the characteristic field C2.This characteristic field can also lead to a rarefaction wave, depending on theinitial data.

The third characteristic field C3 can lead to either to another shock or ararefaction wave. When the flow is such that c > u the eigenvalue λ3(U) isnegative and the characteristics move in the opposite direction of the flow witha rarefaction or expansion wave formed, in which all variables are continuous,and the density of the gas decreases as this wave passes through.

Riemann problem

A Riemann problem consists of an initial value problem with a piecewise con-stant initial data separated by a single discontinuity. When dealing with thesystem of Euler equations, it is also called a shock tube problem as, for aparticular initial condition, it can be idealized experimentally by utilizing a


long tube, initially divided by a diaphragm into two regions, which hold a sta-tionary gas at two different states, left and right, and suddenly removing thediaphragm, allowing the gas to flow. Neglecting any viscous effects the flowcan be considered uniform across the tube and the 1–D Euler equation modelapplies. The Riemann problem for the Euler equations can then be stated as:Find a solution for equation (5.27) with the initial data

U 0 =

UL for x < x0

UR for x > x0

(5.39)

where x0 represents the position of the discontinuity. The structure of theflow for this problem turns out to be very interesting with the typical solutionconsisting of four constant states separated by three elementary waves: a lineardegenerate contact discontinuity wave and two non–linear waves, each of whichmight be either a shock or a rarefaction wave depending on the left and rightinitial states. The three distinct waves, discussed previously, are schematicallypresented in figure 5.6, where a higher pressure on the left region was assumed.The shock wave moves into the region at low pressure, followed by a contactdiscontinuity, and an expansion fan, centred at the diaphragm position, movesin the opposite direction.

WaveRarefaction

ContactDiscontinuity

RightState

LeftState

WaveShockInitial Location

of Diaphragm

t

x

C3 C1 C2

Figure 5.6: Typical solution to the Riemann problem for the 1–D Euler equa-tions.

Apart from the possibility of being realized experimentally, the shock tubeproblem can also be solved semi–analytically with unique solution U ⋄, as longas a vacuum does not form [20, 50, 62]. The solution depends only on x/t


and the initial data UL, UR. Furthermore, it constitutes a significant testcase for the validation of any numerical algorithm developed for the solutionof inviscid compressible flows, and as will be seen later in this chapter, it alsohas a theoretical importance, as many numerical schemes developed to solvethe Euler equations have as their base the successful solution of a Riemannproblem.

5.3 Preliminaries on Numerical Methods

5.3.1 Basic principles of upwind schemes

Upwind schemes attempt to discretise the hyperbolic partial differential equa-tions by introducing the physical properties of propagation into the numeri-cal formulation. First–order upwinding schemes share most of the desirableproperties of monotone schemes, 6.2.3, for the calculation of discontinuous so-lutions, with smooth transitions near discontinuities, and represent the basisfor the development of many high–resolutions schemes utilized for practicalapplications. The propagation of information along characteristics is easilyidentified when a one–dimensional model is considered, as already discussed.Hence upwind schemes can be constructed in a quite straightforward manner.

Linear scalar equations

An attempt to built a numerical scheme using differences biased in the directiondetermined by the sign of the characteristic speed goes back to the Courant–Isaacson–Rees scheme [8], who applied this concept in the discretisation ofthe linear scalar equation of the type given in (5.1). This simple first–orderaccurate upwind scheme, using an explicit time discretisation, can be writtenas

un+1I = un

I − a∆t

∆x

unI+1 − un

I for a < 0

unI − un

I−1 for a > 0(5.40)

where the nomenclature adopted on the spatial discretisation follows the con-ventional finite difference notation. By introducing the wave speeds

a− = min(a, 0) =1

2(a − |a|)

a+ = max(a, 0) =1

2(a + |a|)

(5.41)


equation (5.40) can be rewritten as

un+1I = un

I − ∆t

∆x

[a+(un

I − unI−1) + a−(un

I+1 − unI )]

(5.42)

or, in terms of the flux F = au,

un+1I = un

I − ∆t

∆x

[∆F+

I−1/2 + ∆F−I+1/2

](5.43)

where ∆F±I+1/2 can be seen as the contribution to the flux difference ∆FI+1/2

due to the right and left running wave at the interface I + 1/2, respectively.The dependence on the sign of the characteristic speed can be easily recognizedin equation (5.42) or (5.43). Alternatively, equation (5.42) can be written as

un+1I = un

I − ∆t

2∆x[

CD︷︸︸︷a(un

I+1 − unI−1)−

AD︷︸︸︷|a|(un

I+1 − 2unI + un

I−1) ] (5.44)

where the second term on the right hand side represents a central difference(CD) discretisation of the scalar equation and the third term represents thenumerical viscosity or artificial diffusion (AD) introduced by the upstreamdifference adopted. The last form presents two options to deal with the propa-gation term in a hyperbolic equation in order to have a stable discrete scheme,either to use an upwind–biased discretisation or to adopt a central discretisa-tion and add an artificial viscosity term similar to the one present in (5.44).

The CFL condition

The utilization of a von Neumann stability analysis on the discretised equation(5.44) shows that a necessary condition to guarantee stability of the solutionof a three–point explicit scheme for a hyperbolic linear scalar equation, with aconstant ratio ∆t/∆x, is obtained if

|C| = |a|∆t

∆x≤ 1 (5.45)

where C is called the Courant number and this condition is referred to as theCourant–Friedrichs–Lewy or CFL condition which was introduced by Courantet al [7]. This stability condition implies that the domain of dependence of thediscretised equation for any point P should include the domain of dependenceof the differential equation, which influences the behavior of the system atthis point. Geometrically this can be represented as shown in figure 5.7, andindicates that any disturbance which propagates at speed a should not travel


a distance |a|∆t bigger than the spatial discrete spacing ∆x. However, thiscondition is not sufficient for stability, and for instance it is easily shown [19]that the three–point explicit centered difference, represented in equation (5.44)when the AD term is zero, is unconditionally unstable even if the condition(5.45) holds.

t

∆ t

∆x ∆x

∆ ta ∆ ta

P

x

n+1

n

dtdx

= a+dtdx

= a-

Continuous domainof dependence

Discrete domainof dependence

I

I+1I-1

Figure 5.7: Discrete domain of dependence of a three–point discrete schemeand characteristic propagation for a scalar linear convection equation.

Extension for linear system of equations

The scalar form of the first–order upwind scheme previously described in (5.44)can be applied to each of the decoupled scalar characteristic equations (5.25)leading to the scheme, in matrix form,

Wn+1I = Wn

I − ∆t

2∆x

[Λ(Wn

I+1 − WnI−1) − |Λ|(Wn

I+1 − 2WnI + Wn

I−1)]

(5.46)

which consists of m equations and |Λ| represents the diagonal matrix whichcomponents are the absolute value of the eigenvalues of A. Using the relation(5.26) this scheme can be transformed back into the original variables, takingthe form

Un+1I = Un

I − ∆t

2∆x

[A(Un

I+1 − UnI−1) − |A|(Un

I+1 − 2UnI + Un

I−1)]

(5.47)


and the necessary stability condition (5.45) becomes

|C| = max|λk|∆t

∆x≤ 1 (5.48)

assuming that all equations are discretised with the same time interval ∆t andspacing ∆x.

5.3.2 Non–linear scalar equations

When attempting to extend a discrete scheme, developed for linear equations,to solve numerically a non–linear conservation law some additional difficultiescan arise, such as non–linear instability of the method, non satisfaction of thediscrete conservation property and the possibility of changes in the sign of thejacobian a(u) = ∂F/∂u.

Conservative schemes

Consider the first–order accurate explicit upwind scheme (5.42), with minormodification and assuming un

I ≥ 0 ∀ j, n, it can be straightforwardly extendedto deal with the Burgers equation (5.6) in a quasi–linear form, leading to

un+1I = un

I − ∆t

∆x

[un

I (unI − un

I−1)]

(5.49)

Instead of the form (5.6), let’s consider the Burgers equation in conserva-tion form (5.9). The upwind scheme, assuming again un

I ≥ 0 ∀ j, n, can thenbe defined as

un+1I = un

I − ∆t

∆x

[F (un

I ) − F (unI−1)

](5.50)

which is equivalent to

un+1I = un

I − ∆t

∆x

[a(uI−1,uI)

︷︸︸︷(un

I + unI−1)

2(un

I − unI−1)

](5.51)

where the propagation velocity a(uI−1, uI) is taken as the simple average be-tween un

I and unI−1 instead of the upstream value un

I taken in (5.49). These twoextensions of the first–order upwind scheme have completely different behaviorand the solution, at certain time, using both schemes can be seen in figure 5.8,where the initial data consists of a discontinuous distribution with uL > uR.


(a) (b)

0.8

1

1.2

1.4

1.6

1.8

2

2.2

-1.5 -1 -0.5 0 0.5 1 1.5X

u(x,0)u(x,t)

Anal. u

0.8

1

1.2

1.4

1.6

1.8

2

2.2

-1.5 -1 -0.5 0 0.5 1 1.5X

u(x,0)u(x,t)

Anal. u

Figure 5.8: First–order upwind solutions to Burgers equation. (a) Non–conservative method and (b) conservative method.

The solution in figure 5.8(a), using formulation (5.49), despite looking reason-able, propagates the discontinuity at totally wrong speed, and does not evenrepresent a weak solution of the original problem [25] . In figure 5.8(b) thesolution computed with the use of the method (5.50) has the shock at thecorrect location. This is a direct consequence of Lax–Wendroff theorem [23]which can be stated as:

“If the numerical solution u(x, t) computed with a consistent andconservative method converges, boundedly almost everywhere, toa function v(x, t) as ∆x and ∆t approach zero, then v(x, t) is aweak solution of the conservation law.”

A proof of this theorem can be found in the original paper of Lax and Wendroff[23] or in references [25, 51]. The theorem does not guarantee that the solutionconverges, also, if there exists more than one weak solution, it might be thatthe solution converges to the wrong one and the entropy condition must beimposed. However, it ensures that a solution computed with a conservativemethod satisfies the weak form of the problem, and so the Rankine–Hugoniotcondition at the discontinuities.

The discretised version described in equation (5.50) is said to have theconservative property [2, 25] as it maintains the physical conservation statementexactly (except for round–off errors) for any discretisation over an arbitraryfinite region, with the quantity of a conserved variable changing only due tothe flow through the boundaries. Some non–conservative schemes can stillsuccessfully address hyperbolic equations [11, 31], but they will require someform of shock fitting otherwise the strength and location of the discontinuitiesare not correctly captured [20].


Numerical flux

A general three–point explicit difference scheme in conservation form can beexpressed as

Un+1I = Un

I − ∆t

∆x

FFF I+1/2 − FFF I−1/2

n(5.52)

where FFF I+1/2 is commonly referred to as a numerical flux function and expres-sion (5.52) represents the discrete analogue of a conservation statement. Thenumerical flux function is defined by

FFF I+1/2 = FFF I+1/2(UnI , Un

I+1) with FFF I+1/2(UnI , Un

I ) = F (UnI ) (5.53)

and the second relation above means that the numerical flux function reducesto the true flux in a constant flow and this condition is required in order to beconsistent with the conservation law.

The upwind scheme described in equation (5.50) when written in the form(5.52) has the flux function

FI+1/2 = F (unI ) =

1

2(un

I )2 (5.54)

and is stable only for unI ≥ 0 ∀ j, n. Of course, in a general situation the

Jacobian a(u) = ∂F/∂u can change sign and a switch in the direction of thediscretisation, such as that given in equations (5.41–5.42) or (5.44), must beintroduced. This leads to a direct extension of the linear scheme (5.44) withnumerical flux

FI+1/2 =1

2[ F (uI+1) + F (uI) ] − |a(uI , uI+1)|(uI+1 − uI)n (5.55)

where a is no longer constant, as it is a function of uI and uI+1.

Upwind schemes (Burgers equation)

The expression given in (5.55) is not unique as it is not clear at which point theJacobian a(uI , uI+1) has to be evaluated. Indeed, the numerical flux functionof the non–linear upstream difference schemes, can be cast in the more generalform

FI+1/2 =1

2[ F (uI+1) + F (uI) ] − d(uI , uI+1)n (5.56)


with d(uI , uI) = 0 due to the consistency requirement (5.53) and d(uI , uI+1)represents the dissipation term which can be constructed in numerous ways[18]. In particular, the form described in equation (5.55) has

d(uI , uI+1) = |a(uI , uI+1)|(uI+1 − uI) (5.57)

such that a(uI , uI) = a(uI). There are various ways to compute a(uI , uI+1),and the simplest consists of taking

a(uI , uI+1) = a(

uI + uI+1

2

)(5.58)

Another form has been designed by Roe [44], where the approximate Ja-cobian a(uI , uI) is established such as to satisfy

a(uI , uI+1) =F (uI+1) − F (uI)

(uI+1 − uI)(5.59)

which represents a linearized discrete counterpart of a(u) = ∂F/∂u. For Burg-ers scalar equation, where F (u) = u2/2, a(uI , uI+1) is uniquely determined,and where the expressions (5.58) and (5.59) turn out to determine the samea(uI , uI+1), leading to a scheme which is identical to the Murman–Cole scheme[32] and also the Huang scheme [21]. It should be observed that where a issmall, in particular where a = 0, a problem of indeterminacy exist and, asthe artificial viscosity in (5.47) goes to zero, it enables the scheme to resolvestationary or near–stationary shocks but may also result in admitting non–physical discontinuities, with the necessity of some form of numerical entropycondition to be imposed [18], which will be further discussed later. But the pos-sibility of constructing a linearization similar to (5.59) for non–linear systemsof equations [44] leads to a remarkable generalization of the linear first–orderupwind scheme (5.47), also to be discussed later in this chapter. The numerical

flux for the Roe scheme F (R)I+1/2 can then be written as

F (R)I+1/2 =

1

2

(a+

a

)(uI)

2

2+

(a−

a

)(uI+1)

2

2

n

(5.60)

where the notation defined in equation (5.41) is again used.

A different way to built d(uI , uI+1) was proposed by Engquist and Osher[10] and takes the form

d(uI , uI+1) =∫ uI+1

uI

|a(u)|du (5.61)


For Burgers equation a(u) = u and the integration in (5.61) can be directlyevaluated leading to

d(uI , uI+1) =

(|uI+1|uI+1

)F (uI+1) −

(|uI |uI

)F (uI) (5.62)

which leads to the numerical flux FFF (E−O)I+1/2 for the Engquist and Osher scheme

F (E−O)I+1/2 =

(u+

I )2

2+

(u−I+1)

2

2

n

(5.63)

where the conventions u+, u− follow the same definitions presented in (5.41)and permit the previous compact form for the numerical flux. As a consequenceof the integral in (5.61), the Engquist and Osher numerical flux is differentiableacross a stationary shock.

Godunov scheme (general theory)

Godunov’s method [12], which precedes the previously described schemes, isbuilt up by considering piecewise constant solutions over each mesh cell at afixed time and by solving exactly the Riemann problems at the boundaries be-tween adjacent cells. This was the first–successful conservative upwind schemefor non–linear conservation laws.

Godunov’s method was derived by using the numerical solution un attime–level tn to define a piecewise constant function in x

u(x, tn) = unI for xI−1/2 ≤ x ≤ xI+1/2 (5.64)

which is used as an initial data to define a sequence of local Riemann problemsgoverned by the initial value problem (5.9) with

u0(x) =

uL = unI for x < xI+1/2

uR = unI+1 for x > xI+1/2

(5.65)

Here x measures the distance to the sample interface I + 1/2 and uL, uR arethe right and left interface states. The solution can be expressed exactly usingthe weak similarity solution over a short time interval, i.e. up to the time whenadjacent cells begin to interact, according to

u⋄(x, t) = u⋄(x

t; uL, uR) for xI−1/2 ≤ x ≤ xI+1/2 , tn ≤ t ≤ tn+1 (5.66)


which is a function only of x/t and the initial data uL, uR. For the Burgersequation the solution consists of two constant states separated either by ashock or an expansion fan and is given in [20, 56]. A new piecewise constantapproximate solution u(x, tn+1) is obtained by averaging the cell exact solutionat time tn+1 according to

un+1I = u(x, tn+1) =

1

∆x

∫ xI+1/2

xI−1/2

u⋄(x, tn+1)dx (5.67)

Although, it is not typically implemented in this way, the basic procedureof Godunov’s approach can be summarized as consisting of the following threesteps:

Step1: Definition of a sequence of Riemann problems at cell

interfaces, (5.64) and (5.65);

Step2: Solution of locally independent Riemann problems, (5.66);

Step3: Averaging of interface states at time–level tn+1, (5.67).

and the process repeats up to the desired time–level T .

The weak conservation form of the problem, after solving the local Rie-mann problem, can be written over the rectangle (xI−1/2, xI+1/2) × (tn, tn+1),using equation (5.12), as

∫ xI+1/2

xI−1/2

u⋄(x, tn+1)dx =∫ xI+1/2

xI−1/2

u⋄(x, tn)dx

−∫ tn+1

tnF [u⋄(xI+1/2, t)] − F [u⋄(xI−1/2, t)]dt

(5.68)

which is possible as u⋄(x, t) is assumed to be an exact weak solution.

Note that u⋄(x, tn) = unI as defined by equation (5.64), and that the exact

solution defined by (5.66) is constant at the cell interface, x/t = 0, over theinterval (tn, tn+1) and so is F (u⋄). These facts and the use of equation (5.67)leads to a considerable simplification of the average stage, which then takesthe form

un+1I = un

I − ∆t

∆x

[F (u⋄

I+1/2) − F (u⋄I−1/2)

]n(5.69)

This scheme is clearly in conservation form and the numerical flux is


F (G)I+1/2 =

F (u⋄

I+1/2)n

(5.70)

which is equal to the physical flux F computed for the exact Riemann solutionu⋄ at the interface I + 1/2. The stability analysis of (5.69) requires that

|C| = |a(u)|∆t

∆x≤ 1 (5.71)

This represents a natural generalization of the CFL condition (5.45) for non–linear equations, with the implication that interaction of waves from neighbor-ing Riemann problems is allowed, for C ≥ 0.5, provided it is entirely containedwithin a mesh cell. This fact would made it difficult, or even impossible, tocompute u⋄(x, t). However, equation (5.69) only requires us to compute u⋄ atthe interfaces of the cells, where it remains constant (x/t = 0) and easy to

compute. The resultant numerical flux FFF (G)I+1/2, when written using similar a

notation to (5.63) and the similarity exact solution [20, 56], is

F (G)I+1/2 = max

(u+

I )2

2,(u−

I+1)2

2

n

(5.72)

In the light of Godunov procedure to determine an upwind scheme, Roeand Osher schemes can be seen as different ways of approximately resolvingthe local Riemann problems, with the other steps remaining the same as inGodunov approach.

When the flux F is a convex function of u, i.e. ∂F/∂u > 0, such as inBurgers equation, a unique critical or ‘sonic’ value us is obtained when thewave speed vanishes a(u) = 0 . In analogy to aerodynamics, positive valuesof u (u > us) are assumed as ‘supersonic’ and negative values u (u < us) as‘subsonic’.

If the Roe numerical flux F (R)I+1/2 (5.60) is carefully compared with Go-

dunov numerical flux F (G)I+1/2 (5.72), for each possible situation of the initial

data (5.65), it can be observed that they differ only in the case of an expan-sion with sonic transition. In the Roe scheme, the artificial diffusion termd(uI , uI+1) is identically zero, regardless of the entropy condition, allowingexact resolution of stationary shocks but also admitting nonphysical expan-sion shocks. On the other hand, when analysing the numerical flux of theEngquist–Osher scheme F (E−O)

I+1/2 (5.63) it is noticeable that the scheme detectsthe presence of the stationary point through the evaluation of the sign of u inequation (5.62) and d(uI , uI+1) is no longer null at a such point, being enoughto enforce the selection of the correct expansion shock. The penalty for theextra diffusion incorporated in the Engquist–Osher scheme appears when an


(a) (b)

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5X

u(x,0)Anal. u(x,t)u

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5X

u(x,0)Anal. u(x,t)u

(c)

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5X

u(x,0)Anal. u(x,t)u

Figure 5.9: First–order upwind solutions to Burgers equation using differentRiemann solvers. (a) Roe scheme, (b) Osher scheme and (c) Godunov scheme.

stationary shock is present in the solution, which can no longer be consideredas a discontinuous transition requiring at least one to two intermediate states.

All the previously discussed features can be observed in figure 5.9, wherethe solution of Burgers equation for the propagation of an initial square waveis considered. Both the Roe and the Godunov schemes resolve exactly thestationary shock while the Engquist–Osher scheme represent the shock withtwo intermediate states. The initial stationary expansion shock is broken outby Engquist–Osher and Godunov schemes leading to a stationary expansion fanbut not recognized by Roe scheme in which the entropy–violating stationaryexpansion shock remains a stationary solution at any time. It is also observedthat both Engquist–Osher and Godunov results still present a small unphysicalrarefaction shock, at the sonic point, embedded in the centered expansion fan.This feature is well known [9, 60, 63] and will be investigated in more detailwhen dealing with Euler equations.


5.4 1–D Euler System of Equations

As already mentioned, for a system of equations the eigenvalues of the Jacobianmatrix A can in general have different signs with waves propagating in bothdirections. For a linear system of equations this represents no problem as theeigenvalues are constant and a straight generalization of the scalar upwindscheme is possible yielding the formulation (5.47). For a non–linear system ofequations, this approach does not work directly as the Jacobian matrix is nolonger constant, with not a single definition for A(UI , UI+1).

There are basically two large families of techniques to extend upstreamdiscretisation in a conservative form for a system of non–linear equations. Thefirst approach consists in a generalization of the Godunov procedure, describedin section 5.3.2, in which a local characteristic structure is obtained by solvingin different ways Riemann problems at the interfaces of the discrete cells. Theseschemes are normally referred to as Godunov–type schemes or flux differencesplitting methods, and they distinguish between the influence of the forward–and backward–moving waves. The second approach is achieved by splittingthe flux term in parts, each of which having an unambiguous direction ofupwinding determined by the sign of the associated propagation speed. Theschemes which adopt this strategy are called flux vector splitting methods,and they distinguish between the influence of the forward– and backward–moving particles. A third class of methods can be devised to represent schemeswhich are either a mixture of the two mentioned approaches or which can notformally be incorporated into one of these two classical approaches, and will begenerically referred to here as flux splittings. In a general sense, these schemesdiffer with respect to the level of introduction of physical information in thediscretisation process, giving rise to different levels of accuracy and robustnesswhen solving the Euler equations.

Although, all upstream difference schemes appear to comprise a largeamount of artificial diffusion the biggest distinction among them occurs whenthe characteristic speed is close to zero or zero and this shows up in the wayin which each scheme resolves a stationary or slowly moving shock, centeredtransonic rarefaction fan or contact discontinuity. Moreover, when extendingthe upwind discretisation to deal with the convective term of the Navier–Stokesequations the dissipative properties near contact discontinuities and slip sur-faces are also quite distinct with serious consequences on the overall accuracyof the simulated results. Finally, as practical higher–resolution formulation in-herit the properties of the scheme from which it originates the various upwindapproaches will be discussed next.


5.4.1 Flux difference splitting approach

For a scalar non–linear conservation law, the adoption of any of the schemesdescribed in section 5.3.2 does not represent a crucial difference in terms ofoperation count (and CPU time), and there is no reason not to adopt theGodunov scheme with exact Riemann solver. But when a system of conser-vation laws is considered, this is not the case and the operation count willbe very different, depending on the scheme adopted. This is even more pro-nounced for multi–dimensional simulation or extension for real gas computa-tion [29, 59, 64]. One difficulty with Godunov’s scheme refers to its complexity,as it is often difficult or impossible to find a closed solution for the Riemannproblem. This leads, in general, to a set of non–linear algebraic equationswhich must be solved iteratively, making the scheme less attractive due to theassociated computational cost. It is also evident that due to the averaging ofthe exact solution over the cells (5.67), a large amount of numerical error isintroduced. Therefore, it is not worthwhile to evaluate the Riemann solutionexactly and it might be possible to achieve equally good numerical results withan approximate Riemann solver which gives an explicit expression for the fluxfunction, requiring less computational work and with an accuracy compatibleto the overall accuracy of the final scheme. Godunov recognized this problem[13] and simplified the scheme by using an approximate solution to the exactRiemann solution, where only shocks were considered which represents a dan-gerous procedure as expansion shocks may show up [38]. Another approach,normally preferred, consists in first obtaining a solution to an approximateRiemann problem, and then adopting it as the desired solution to the originalRiemann problem. The approximate Riemann problem is presumably easierto solve and must retain the main properties of the original problem.

The form presented in equation (5.43) to express the upwind discreteformulation for a scalar equation gives rise to the term flux difference splitting([44]), which refers to all upwind schemes that can be cast in this form. Forthe system of equations the corresponding numerical flux can be written as

FFF I+1/2 =1

2

(F I + F I+1) − (∆F

+I+1/2 − ∆F

−

I+1/2)n

(5.73)

where the flux difference

∆F I+1/2 = F I+1 − F I = ∆F+I+1/2 + ∆F

−

I+1/2 (5.74)

has been splitted with

∆F+I+1/2 = −( FFF I+1/2 − F I+1) ∆F

−

I+1/2 = ( FFF I+1/2 − F I) (5.75)


and ∆F+I+1/2, ∆F

−

I+1/2 denote the contribution to the flux difference arisingfrom positive and negative waves, respectively.

A large number of schemes are included in this family, such as the Go-dunov scheme [12] , the Roe scheme [44], the Osher scheme [34], the Harten,Lax and van Leer scheme [18], Einfeldt’s HLLE scheme [9] and the Pandolfischeme [39]. Within these schemes the most popular are the Roe and theOsher schemes which are now discussed in detail, applied to the 1–D Eulerequations.

Roe flux difference splitting

Roe [44] developed an approximate Riemann solver by computing the exactsolution to a modified conservation law with the flux F (U) = AU, i.e. con-sidering the local linearized system of equations

∂U

∂t+ A(UL, UR)

∂U

∂x= 0 (5.76)

with the initial data

U 0(x) =

UL = UnI for x < xI+1/2

UR = UnI+1 for x > xI+1/2

(5.77)

This linear Riemann problem is relatively easy to solve explicitly [25] byexpressing the variations δU of the conservative variables as a sum of thesimple wave contributions as described in equation (5.24). But the problemof determining the constant matrix A(UI , UI+1) in a reasonable way persists.Roe defined A(UI , UI+1) as a mean value of the Jacobian matrix A(U) =∂F /∂U such that it satisfies the following requirements:

(R1) A(UI , UI+1) → A(U) for UI , UI+1 → U;

(R2) A(UI , UI+1) is diagonalizable with real eigenvalues;

(R3) A(UI , UI+1)∆UI+1/2 = ∆F I+1/2 for any pair UI , UI+1.

(5.78)

Condition (R1) guarantees that the linearized matrix A(UI , UI+1) is con-sistent with the linear system (5.76). Condition (R2) requires that the systemof equations (5.76) must be hyperbolic. Condition (R3) ensures conservationfor the final scheme and so the satisfaction of the Rankine–Hugoniot condi-tion (5.16). Hence the approximate Riemann solution agrees with the exact


Riemann solution when the left and right states are connected by a singlediscontinuity. As a consequence, a crisp representation, with at most a sin-gle interior state is obtained which also does not allow for the spreading ofexpansion waves leading to non–physical expansion shocks.

The derivation of a mean matrix for a general system which satisfies allthe mentioned requirements (5.78) is not possible. Fortunately, for specialsystems of equations, such as the Euler equations for a perfect gas, it is notonly possible but there is also an unique derivation of the matrix A [47]. Roeobserved that the vectors U and F are merely quadratic functions of theparametric vector ω = ρ1/2(1, u, H)T , with H = (ε + p/ρ) being the totalenthalpy. This allows a straightforward calculation of the matrix A. Theresultant matrix A is identical to the local Jacobian matrix given by (5.29),when expressed as a function of ρ, u and H , if these variables are replaced byan average weighted by the square root of the densities. The interface statesare then defined according to

ρI+1/2 =√

ρIρI+1

uI+1/2 =(u√

ρ)I + (u√

ρ)I+1√ρI +

√ρI+1

HI+1/2 =(H

√ρ)I + (H

√ρ)I+1√

ρI +√

ρI+1

(5.79)

Utilizing the condition (R3) in (5.78) Roe split the flux difference ∆F I+1/2

according to its right ∆F+I+1/2 and left ∆F

−

I+1/2 moving parts as

∆F I+1/2 = ∆F+I+1/2 + ∆F

−

I+1/2 = A+∆UI+1/2 + A

−∆UI+1/2 (5.80)

with A+

and A−

defined as for the scalar equation (5.41), i.e.

A+

=A + |A|

2A

−=

A − |A|2

(5.81)

This allows us to write the resulting Roe’s interface numerical flux as

FFF (R)I+1/2 =

1

2

[ F (UI+1) + F (UI) ] − |AI+1/2|(UI+1 − UI)

n(5.82)

Here |AI+1/2| can be computed using the transformation (5.22) according to

A = R|Λ|R−1 (5.83)


where the eigenvalues and matrix of the eigenvectors Λ, R and R−1

of the lin-earized matrix A can be computed as described in [20, 44], using the interfacestates defined in (5.79). The numerical flux (5.82), using the decomposition ofthe variation of the conservative variables δU into the variation of the charac-teristic variables δWK given by (5.24), can also be written as

FFF (R)I+1/2 =

1

2

[ F (UI+1) + F (UI) ] −

m∑

k=1

rk|λk|δWk

n

(5.84)

As the absolute value of the matrix A appears in (5.82) multiplied onlyby the vector of the variation of U, there is no need to compute explicitly R,

Λ and R−1

and an alternative, computationally more efficient, way to obtainthe dissipation vector DI+1/2 = |AI+1/2|∆UI+1/2 was derived by Turkel [54].This is presented in appendix B.

The more relevant achievements of Roe’s approach relate to the computa-tional efficiency of the approximate Riemann solver when compared with theexact solver and the property of allowing an exact representation of a singlediscontinuity. The main drawbacks of the method, resulting from the lineariza-tion adopted, refers to the fact that it does not recognize stationary expansionwaves, which are not allow to spread, and the presence of some anomaliesdeveloped in strong bow shock formation [41].

Entropy fix

Different ways to remedy the entropy violating problem are available [17, 35,46, 56]. Harten and Hyman [17] suggest to replace |λk| in the computation of|A| by a continuous function Ψ(λk), for which the standard expression is

Ψ(λk) =

|λk| |λk| ≥ δk

(λk2+ δ2

k)2δ2

k

|λk| < δk

(5.85)

where δk denote small positive numbers, either constant and the same for allfields “k” or more elaborately defined as a function of the local flow condition[20, 64].

Osher flux difference splitting

Osher [34, 38] extended to hyperbolic systems of conservation laws the schemedeveloped by Engquist and Osher [10] for a single conservation law. Osher’sapproach to solve the Riemann problem considers the flux difference betweenthe left and right states according to


∆F I+1/2 =∫ UI+1

UI

A(U)dU (5.86)

Utilizing a similar splitting of the flux difference as described in equations(5.80) and (5.81), but now utilizing the flux difference defined according to(5.86), Osher’s numerical flux can be expressed as

FFF (O)I+1/2 =

1

2

[ F (UI+1) + F (UI) ] −

∫ UI+1

UI

|A(U)|dU

n

(5.87)

The integration can be evaluated along an arbitrary path in phase space fromUI to UI+1. Osher chose a path that consists of m simple wave subpathsassociated with the eigenvalues λk and eigenvectors rk. Using the relation(5.24) in (5.87), the numerical flux becomes

FFF (O)I+1/2 =

1

2

[ F (UI+1) + F (UI) ] −

m∑

k=1

∫

Γk

rk|λk|dW

n

(5.88)

where Γk is the subpath along the simple wave k. Since the integration pathsfollow simple wave solutions, the approximate Riemann solution involved inthis decomposition depends only on characteristics and hence does not containdiscontinuous transitions [20]. The evaluation of the integral in (5.88) requiresonly the knowledge of the intermediate states which connect two subpathsΓk and Γk + 1 and of any sonic states which might occur along the pathsassociated with the genuinely nonlinear fields (see figure 5.10).

For the Euler equations, two intermediate states are necessary and, byusing the constancy of the characteristic variables δWk through a simple wavepath, Osher deduced explicitly both intermediate and possible sonic states. Aprecise derivation and description of these formulas are presented in [5, 36].The choice of the path and the order for their computation is not unique, andindeed the final result of the integral in (5.88) is path–dependent [5]. Osherand Solomon [38] select the order of decreasing eigenvalues, which enablesthe proof that the resultant numerical scheme always give monotone solutionsthrough a steady shock and that the limiting solution satisfies the entropycondition. Furthermore, Roberts [43] shows that this order represents the bestchoice when slowly moving shock waves are considered, with basically no noiseradiating from the shock.

The main properties of Osher’s scheme are related to its smoothness, andso differentiability which might be an important issue for implicit formulations,and the automatic satisfaction of the entropy condition. However, it does notrepresent a cheap alternative to the exact Riemann solver and an extension toequilibrium real gases is not known at this point.


u-cλ=u+cλ=

uλ=Γ1

Γ2

Γ3

I

U1

U2

Possible Sonic Points

Intermediate Points

Discrete PointsI+1

I+1/3 I+2/3

I1∗

I3∗

Figure 5.10: Integration path in phase space for Osher’s scheme.

5.4.2 Flux vector splitting approach

The simplest way of introducing upwinding into system of hyperbolic lawsseems to be based on the representation of the flux vector as

F (U) = F+(U) + F

−(U) (5.89)

where the Jacobian matrices B−(U) = ∂F−/∂U and B+(U) = ∂F

+/∂U have

no positive and no negative eigenvalues respectively. This definition of F+

andF

−enables the system of hyperbolic conservation laws to be written according

to

∂U

∂t+

∂F+

∂x+

∂F−

∂x= 0 (5.90)

An upwind scheme, in conservation form, is obtained by applying backwardand forward differencing to ∂F

+/∂x and ∂F

−/∂x respectively. This idea

seems to have been first used in astrophysical gas calculations [48] and tohave been rediscovered in a different context by Steger and Warming [53].Referring to the Euler equations, this procedure can be interpreted as a naturalconsequence of regarding a fluid as an ensemble of particles, where some ofthem will move forward and others backward [18]. Harten et al [18] showthat this approach can be derived by approximating each conservation lawin Euler’s equations by the collisionless Boltzmann equations and suggeststhe term Boltzmann–type schemes for this family of schemes. This approachclearly does not represent a realistic description of a continuum fluid, but yetthe model seems to work almost as well as the FDS approach, which more


plausibly infer the passage of waves from a comparison of the neighbouringstates [45].

The flux vector splitting approach can also be considered as another wayto merge the two state vectors on the left and right side of the cell interface,leading to the numerical flux

FFF I+1/2 = F+(UI) + F

−(UI+1) (5.91)

or

FFF I+1/2 =1

2

[ F (UI+1) + F (UI) ] − [ ∆I+1/2F

+ − ∆I+1/2F−

]n

(5.92)

with ∆I+1/2F±

= (F±

I+1 − F±

I ). This represents an alternative to be used inthe place of an approximate Riemann solver.

Although the ultimate goal of this research work is the solution of viscousflows, for which the higher resolution schemes based on flux vector splittinghave been proved to give artificially broader boundary layers and inaccuratetemperature distribution on solid bodies [58], the knowledge of the conceptsinvolved in such approaches through the study of classical schemes, such asSteger–Warming [53] and Van leer [55], is very important as it represents thebasis for some modified flux splitting schemes which possess some promisingfeatures. These will be discussed later in this chapter.

Steger and Warming flux vector splitting

Using the fact that the Euler equations in gasdynamics, considering a perfectgas, satisfy the homogeneous property (5.31) Steger and Warming [53] definedthe split flux according to

F+

= A+U and F−

= A−U (5.93)

with A+ and A− defined by (5.81) and corresponding to the Jacobians asso-ciated with the negative and positive eigenvalues λk of the Jacobian matrix A.Using the transformations given in (5.22), it follows that

Λ = Λ+ + Λ− and |Λ| = Λ+ − Λ− (5.94)

or

λk = λ+k + λ−

k and |λk| = λ+k − λ−

k (5.95)


where λk is such that if λk > 0, then λ+k = λk and λ−

k = 0, with the converseresult for λk < 0. Using the above definitions, the matrices A+, A− canbe computed and also the fluxes F

+and F

−. A general expression for the

evaluation of the flux vectors F±

for any eigenvalue splitting which satisfies(5.95) is given in [20, 53],

The resultant scheme has the numerical flux given by (5.91) and representsa very efficient computational scheme. However, it gives very poor results forstationary contact discontinuities, where the dissipation does not vanish, andthe presence of oscillations at sonic points, due to the lack of differentiabilityof the flux split at this point. The cure for the last mentioned problem issomewhat similar to that presented for Roe scheme and was proposed by Steger[52] by redefining λ±

k as

λ±k =

λk ± (λ2k + δk)

1/2

2(5.96)

with δk being an small parameter.

Van Leer flux vector splitting

Van Leer [55] devised an alternative flux vector scheme, without reference tothe homogeneous property of the fluxes in Euler equations, by imposing acertain number of constraints on the definition of F

+and F

−. The main

design features of Van Leer splitting are

(VL1) The split fluxes F±are continuously differentiable;

(VL2) The eigenvalues of the Jacobian matrix B+must be

positive or zero and those of B−negative or zero;

(VL3) For subsonic regimes one eigenvalue is equal to zero;

(VL4) F±

= f(M), i.e. are functions of the Mach number,

and they must be polynomials of the lowest possible

degree.

(5.97)

Here, B± = ∂F±/∂U and it should be noted that, in general, B± 6= A±

[20, 53]. These conditions ensure noticeably better results around sonic pointsand sharper stationary shock transitions, with no more than two intermediatezones, just as the Osher scheme. This leads to an overall better performancewhen compared with Steger and Warming flux vector splitting [3]. It also


retains the computational efficiency of Steger and Warming scheme and rep-resents a differentiable flux splitting formula. However, for stationary contactdiscontinuity, with or without slip, |B| has no vanishing eigenvalue, leadingto very poor resolution and once again disqualifying its use for Navier–Stokessimulations [46].

5.4.3 Other flux splitting schemes

The computational efficiency, and the simplicity, of the flux vector splittingapproach has driven much research in attempting to minimize the intrinsicproblems in resolving accurately stationary discontinuities and slip surfaces,which lead to disastrous results in viscous simulation. Hanel et al [16] proposeda modification to the Van Leer scheme in which the energy–flux splitting issuch that the resulting scheme preserves the total enthalpy. Apart from smallimprovements for low flow speeds reported in [16], the improvements in thehypersonic regime is insignificant [57] and the original problems of the Van Leerscheme persist. Liou et al [29] show that by relaxing the Van Leer conditions,

to define F±, a broader family of continuously differentiable split fluxes can

be generated. They also realize that the pressure splitting could be consideredseparately, allowing extra flexibility in the search for new schemes. Withoutdeparting from the idea of a successful pure flux vector splitting scheme, Liouand Steffen [27] and Coirier and Van Leer [6] tried the use of higher–orderpolynomial expansions for the mass flux and different pressure expansions, buttheir scheme seems not to be robust enough in multi–dimension calculations[60].

Another effort to develop less-dissipative schemes, combines flux differ-ence and flux vector splitting ideas and was first suggested by Hanel et al [15]who substitute the transverse momentum flux by one borrowed from flux dif-ference splitting. Despite some progress on boundary layer thickness, it doesnot improve the wall temperature prediction, it introduces pressure irregular-ities across the boundary layer and is only directly implemented in terms ofstructured grids, where the streamwise and transverse directions are readilyidentifiable. Inspired by Hanel’s modification, Van Leer [57] extended thishybridization by introducing a similar modification for the energy flux withexpressive enhancement of temperature distribution, but still remaining withthe pressure irregularities.

The improvements achieved with these preliminary attempts of blendingflux vector and flux difference splittings and the hope to obtain a simple, com-putationally efficient, algorithm, with accuracy rivalling flux difference split-ting approaches, led to the recent schemes proposed by Liou and Steffen [28]and the variations designed by Wada and Liou [60].


Liou and Steffen splitting

The flux F in the Euler equations can be regarded as a sum of two physicallydistinct parts, the convective flux F (C) and pressure flux F (P ) according to

F = F (C) + F (P ) = u

Θ︷︸︸︷

ρρuρH

+p

∆︷︸︸︷

010

(5.98)

With this splitting, the Euler equations are expressed as

∂U

∂t+

∂F (C)

∂x+

∂F (P )

∂x= 0 (5.99)

The flux vector splitting numerical flux (5.91) can then be written as

FFF I+1/2 =(uIΘI + pI∆)+ + (uI+1ΘI+1 + pI+1∆)−

n(5.100)

Motivated by the fact that the terms F (C) and F (P ) can be treated sepa-rately and by the idea of mixing flux difference and flux vector splittings Liouand Steffen [28] proposed to replace (5.100) by

FFF I+1/2 =(u+

I+1/2ΘI + p+I ∆) + (u−

I+1/2ΘI+1 + p−I+1∆)n

(5.101)

with

u±I+1/2 =

uI+1/2 ± |uI+1/2|2

(5.102)

where uI+1/2 can be considered as a suitably defined interface velocity, givenby

u = uI+1/2 = f(uI , uI+1) = (uI)+ + (uI+1)

− (5.103)

An interface pressure, pI+1/2, can be devised as

p = pI+1/2 = f(pI , pI+1) = p+I + p−I+1 (5.104)

The hybrid nature of the numerical flux (5.101) is directly observed as theflux is partially split and partially merged with the utilization of the interfacevalues.


The velocity splitting (uk)± can be evaluated in various ways. Following

Liou [26], the velocity splitting can be defined as

(uk)± =

[(uk) ± |(uk)|]2

if |(uk)| ≥ ck

± [(uk) ± ck]2

4ckotherwise

(5.105)

where ck is the speed of sound at node k.

Different methods of splitting the pressure, as a function of the Mach numberMk, are possible [27]. The simplest choice splits the pressure according to

p±k =

pk[Mk ± |Mk|]2Mk

if |Mk| ≥ 1.0

pk[1 ± Mk]

2otherwise

(5.106)

so that a simple first order expansion of the characteristic speeds [1 ± Mk] isused in the subsonic range and fully upwind for supersonic regime.

The velocity split (5.105) and the pressure split (5.106) were designed torecover the fully upwind scheme in supersonic regions and to account sepa-rately for the convective and acoustic waves. The final scheme was coined byLiou and Steffen [28] as AUSM (Advection Upstream Splitting Method), andsubstituting of the definitions of the interface quantities given in equations(5.102) to (5.104) into equation (5.101), results in

FFF I+1/2 =1

2u[ΘI + ΘI+1] − |u|[ΘI+1 − ΘI ] + 2p∆n (5.107)

which represents a more convenient way to express the AUSM numerical flux.Instead of using an interface convective velocity uI+1/2, an interface Machnumber MI+1/2 can be adopted [28], defined according to

M = MI+1/2 = (uI)+/cI + (uI+1)

−/cI+1 (5.108)

In this case the numerical flux of equation (5.107) is replaced by

FFF I+1/2 =1

2

M [cIΘI + cI+1ΘI+1] − |M |[cI+1ΘI+1 − cIΘI ] + 2p∆

(5.109)


It is the author’s experience [30] that these two different flux formulations leadto similar approximate solutions for the inviscid flow computations tested. Forviscous computations, the split–velocity produces oscillations in the solutionin the vicinity of viscous walls. The explanation for such different behaviormight follow Liou’s [26] argument that the split Mach version introduces extramixing at contact discontinuities and slip lines, across which there may begreat difference in sound speed. For problems with these features present in thesolution, therefore, the use of the split–Mach version is essential if meaningfulresults are to be obtained. The AUSM retains all the positive properties of VanLeer flux vector splitting and it also recognizes contact discontinuities, whereno artificial dissipation is added. Other good properties, such as possessingpositive property, being free from the so called carbuncle phenomena [41] arereported by Liou and Steffen [28].

Equation (5.99) can be re–written in semi–linear form according to

∂U

∂t+ A(C)

∂U

∂x+ A(P )

∂U

∂x= 0 with A(C) + A(P ) = A (5.110)

where A(C), A(P ) denote the Jacobian of the convective flux ∂F (C)/∂x andof the pressure flux ∂F (P )/∂x, respectively. The eigenvalues of the matrixA(C) are [ γu, u, u ], while those of the matrix A(P ) are [ 0, 0,−(γ − 1)u ]. Thisindicates that all information in F (C) convects with the flow while that asso-ciated to F (P ) convects in the opposite direction, which does not reflect thetrue behaviour of the acoustic waves.

Supported by previous physical arguments, Halt and Agarwal [1, 14] sug-gest a modification on the split given in equation (5.98), by taking out thepressure from the convective term in the energy equation. This leads to

F′

(C) = u

Θ′

︷︸︸︷

ρρuρε

F

′

(P ) =

∆′

︷︸︸︷

0ppu

(5.111)

and represents a full split into convective and acoustic waves, being referredto as WPS (Wave/Particle Split). In fact the idea for such a splitting isnot new and has been used, in a different context, to solve the parabolizedNavier–Stokes equations by Schiff and Steger [49], according to reference [53].This splitting was also used by Rao and Deshpande [42] and by Balakrishnanand Deshpande [4] in a similar context as here. The eigenvalues associatedto the matrix A′

(C) are [ u, u, u ] while that connected to the matrix A′(P ) are

[ 0,√

(γ − 1)/γ c,−√

(γ − 1)/γ c ], which indicates a more distinct split of theconvection and acoustic terms.


The steps employed in the AUSM scheme are adopted, with identicaldefinition for uI+1/2, PI+1/2 and

pu = (pu)I+1/2 = f((pu)I , (pu)I+1) = (pu)+I + (pu)−I+1 (5.112)

with (pu)±k defined similarly to p±k (5.106).

In reference [1], Agarwal and Halt present some 2–D subsonic and tran-sonic Euler computations on unstructured grids where the WPS compares fa-vorably to the AUSM scheme. Nevertheless, no evidence has been presented todate on the performance of the method to the supersonic or hypersonic regimesor for viscous computation, where the AUSM has been given remarkably goodresults [28, 30].

Wada and Liou mixed flux vector/difference generalization

Recently Wada and Liou [60] generalized the hybrid FV/FD AUSM schemeand refer to the AUSMD, AUSMV and AUSMDV, where “D”, “V” and “DV”denote the predominant flux difference character, flux vector character andmixture of the previous two schemes, respectively. The fundamental feature isto split the mass flux (ρu)I+1/2 (≡ ρu) instead of the interface velocity uI+1/2.The numerical flux for the AUSMD scheme is written as

FFF I+1/2 =1

2(ρu)[ΨI + ΨI+1] − |(ρu)|[ΨI+1 − ΨI ] + 2p∆n (5.113)

where Ψ = (1, u, H)T . With appropriate choice for the mass flux splitting,equations (5.107) and (5.109) can be recovered as particular cases of theAUSMD scheme (5.113), see [60]. Wada and Liou use the term AUSMVwhen the term (ρu2)I+1/2 in the momentum flux is modified from the orig-inal AUSMD term and defined as

ρu2 = (ρu2)I+1/2 = u+I (ρu)I + u−

I+1/2(ρu)I+1/2 (5.114)

This scheme includes the Van Leer/Hanel scheme [57] for a particular choiceof the mass flux.

Several issues including: the removal of numerical dissipation at contactdiscontinuities, how to blend AUSMD and AUSMV, entropy fix and a cure forthe carbuncle phenomena are addressed in Wada and Liou’s paper [60] andthe interested reader is referred to it for details.


5.5 Compendium of First–Order Upwind Nu-

merical Fluxes

The general expressions for the numerical fluxes using FD, FV and FD/FVsplittings are given in equations (5.73) (5.92) and (5.113) respectively. How-ever, in order to stress the similarities and differences between the schemesimplemented here, a somewhat unified form is used in the summary of numer-ical fluxes presented next. It should be clear that the forms given below donot necessarily represent the one actually implemented nor the most efficientway to do so. Furthermore, all numerical fluxes will consider the notation offigure 5.11, which will be convenient for future extension to multi–dimensionalunstructured discretisations.

II L IIS ISIR

I L I IS IR

IRI+2 =>

ISI+1 =>

II =>I LI-1 =>

Figure 5.11: One–dimensional stencil and cell interfaces.

Apart from the schemes with a split of the convective and the pressureterms, such as Liou–Steffen AUSM in which the numerical dissipation has amore complex structure, the numerical fluxes can be cast as

FFF IIS=

1

2[(uIΘI +uIS

ΘIS)+ (pI + pIS

)∆]− [A(UI , UIS)(UIS

−UI)] (5.115)

where the term inside the first square bracket represents a simple centraldifference discretisation and where for each choice of the dissipation matrixA(UI , UIS

) a novel scheme is determined.

Roe scheme

FFF IIS=

1

2[CD] − [|A|(UIS

− UI)] (5.116)


Osher scheme

FFF IIS=

1

2[CD] − [

∫ UIS

UI

|A(U)|dU] (5.117)

Steger–Warming scheme

FFF IIS=

1

2[CD] − [|AIS

|UIS− |AI |UI ] (5.118)

Van Leer scheme

FFF IIS=

1

2[CD] − [

∫ UIS

UI

|B(U)|dU] (5.119)

Liou–Steffen (AUSM) scheme

FFF IIS=

1

2[M(cIΘI + cIS

ΘIS) + (2p∆)] − [|M |(cIS

ΘIS− cIΘI)] (5.120)

Halt–Agarwal (WPS) scheme

FFF IIS=

1

2[M(cIΘ

′I + cIS

Θ′IS

) + (2∆ ′)] − [|M |(cISΘ′

IS− cIΘ

′I)] (5.121)

with (∆ ′)T = [ 0, p, pu ].

The Roe and the Steger–Warming schemes differ only in the evaluation ofthe absolute Jacobian. The first scheme defines an average state to compute A,which is taken outside the difference operator. If an eigenvalue of A vanishes,the corresponding eigenvalue of the dissipation matrix vanishes too. This isnot true for the eigenvalue of the dissipation matrix corresponding to theSteger–Warming scheme, which does not vanishes in any steady–state andhas discontinuous eigenvalues whenever the corresponding eigenvalues of A(U)vanish [58].

The Van Leer scheme (5.119) is written in a form that resembles Osher’sscheme (5.117) [58], but, since |B(U)| is a perfect gradient the integration is

independent of the path. By the definition of F±(U) in the Van Leer scheme,

one eigenvalue of |B(U)| will vanish somewhere in between a steady stateshock given by UI , UIS

, just as for the Osher’s scheme. However, contrary tothe Osher’s scheme there is no vanishing eigenvalue if UI , UIS

form a steadycontact discontinuity, with or without slip [58]. The Osher’s scheme is normallyclaimed to be more complex to implement than other conventional schemes,


which is in fact not true, however, it does requires more operations than theother schemes analysed here.

The main differences between AUSM or WPS and the other splittings referto the split of the convective and pressure terms a priori to define the numericaldiscretisation and the use of weighted averages to deal with these terms insteadof the simple average given by central difference discretisation. This resultsin a quite elaborate numerical diffusion with contributions added by all threeterms present in the numerical flux (5.120) and (5.121). In the purely diffusiveterm, the third term in equation (5.120) and (5.121), the cell interface Machnumber M is taken outside the difference operator, just as the matrix A ofRoe’s scheme. But in AUSM and WPS it is merely a scalar coefficient |M |,which requires less operations to compute. The cell interface Mach number Mis designed to determine a proper upwind of the convective quantities and alsoto vanish in stationary contact discontinuities. The only difference betweenAUSM and WPS is the way each scheme handles the pressure term in theenergy equation.

Calculation of the time–step

An explicit time integration will be utilized for the applications to bepresented in this chapter. This is obtained by implementing the schemes aswritten according to equation (5.52), where the numerical fluxes are computedat time–level tn. The nodal time–step is taken as the minimum over the time–steps of the adjacent edges, i.e.

∆tI = min[ ∆tILI , ∆tIIS] (5.122)

This represents a local time–step, normally used for steady–state simulations,and where

∆tIIS=

C∆x

max[ (λmax)I , (λmax)IS]

(5.123)

For the Euler system of equations (λmax)I is given by, (5.30),

(λmax)I = |uI | + cI (5.124)

Finally, the global value of the allowable time–step is determined by takingthe minimum nodal time–step of the discretisation

∆t = min[ ∆tI ] for I = 1, . . . , N (5.125)

which is used for transient simulations.


Actually, when Osher’s Riemann solver is adopted, the maximum eigen-values λmax must be taken as the maximum over the subpaths. Referring tofigure 5.10 and considering u > 0, the path Γ1 gives the biggest eigenvaluesand equation (5.123) is replaced by

∆tIIS=

C∆x

max[ (λmax)I , (λmax)I∗1, (λmax)I+1/3 ]

(5.126)

The Courant number C is restricted by the CFL condition given in equa-tion (5.48), i.e. C ≤ 1. This stability limit is not true for the flux vectorsplitting schemes and for Liou–Steffen’s AUSM scheme, as the eigenvalues tobe considered in the calculation of C must be the eigenvalues of |B(U)| and notthe Jacobian eigenvalues of |A(U)|, [20]. Alternatively, a more restricted limit-ing on the Courant number can be obtained for practical purposes [20, 55]. Forthe study of the behavior of the different first–order upwind schemes the time–step used for all numerical applications is computed using equations (5.122) to(5.125). This avoids complication introduced by involving different expressionsand CFL parameters to compute ∆t.

5.6 Numerical Results and Discussion

Several Riemann problems, (5.39), are analysed with the previously describedfirst–order upwind schemes. The weaknesses and strengths of each scheme arestressed throughout this section and the remedies for some well known prob-lems are described. The study of explicit first–order schemes, in space andtime, is essential for the comparison of the performance of the basic schemes,without the uncertainty factors associated to higher–order extensions and im-plicit formulations. Furthermore, this study is important as high–resolutionextensions will, in general, inherit both good and bad features of the first–orderscheme used for their construction.

All the cases analysed employ a spatial grid with 100 discrete points in adomain, which extends from −0.5 to 0.5, and the position of the discontinuityfor the initial data at x0 = 0.0. A CFL number of 0.80 is used and an ideal gaswith γ = 1.4 is considered. The time for stopping the computations (Tmax)has been chosen, for each application, in order to use the full computationaldomain. In the figures, the solid lines represent the semi–analytical solutions[62] to the Riemann problem under consideration.


5.6.1 Shock tube problem: rarefaction wave with a sonic

point

The first test case consists of a Riemann problem in which the left and rightstates are connected by a rarefaction wave [9] only. The left and right statesare defined by

UL = [ ρR (cL/cR)2/(γ−1) , −(cL + cR), ρLc2L/γ ]T for x ≤ 0.0

UR = [ 1.205, 0.0, 10.0 ]T for x > 0.0

with cL =3 − γ

γ + 1cR .

This supersonic shock tube problem (Mmax = 2.5) is studied to check theresolution of an entropy violating stationary shock. The solutions, in terms ofdensity distribution, are presented in figure 5.12 for the time Tmax = 0.05,which corresponds to approximately 51 time–steps.

The bad behavior of the standard Roe scheme and the presence of a smalldiscontinuity at the sonic point when the standard Steger and Warming schemeis used are expected. Also, as already observed in figure 5.9, even schemes suchas the Osher, the Godunov and the Van Leer schemes, which are entropy sat-isfying schemes [37], present an unphysical rarefaction shock embedded in theexpansion fan. According to Woodward and Collela [63], this small rarefac-tion shock occurs because of a subtle property of these schemes related to thefact that the error in the computed flux is one order larger at the cell inter-face, where the flow is sonic, than everywhere else. The current AUSM andWPS schemes also present a small “kink” in the solution, with WPS behavingslightly better. The Osher and Van Leer schemes present almost identical so-lutions. The hybrid FDS/FVS schemes, AUSM and WPS, give a less roundedcorner at the top of the expansion fan. The Steger and Warming and AUSMschemes give the worst behavior at the bottom corner of the expansion fan,with the appearance of a small depression.

To eliminate the glitches at sonic points, Wada and Liou propose to modifythe numerical flux of the scheme whenever a sonic point is found across thecell interface, i.e.

if λ2(UI) < 0 and λ2(UIS) > 0 then

FFF IIS= FFF IIS

− δ′ ∆[ λ2(U) ]IIS× [A.V.]

elseif λ3(UI) < 0 and λ3(UIS) > 0 then

FFF IIS= FFF IIS

− δ′ ∆[ λ3(U) ]IIS× [A.V.]

endif

(5.127)


where δ′ is a constant parameter, ∆[ ]IIS= [ ]IS

− [ ]I and [A.V.] representsthe dissipative term of the scheme.

The solutions obtained using the entropy correction defined in equation(5.85) with δk = 1.0 and using the entropy fix given in (5.127) with δ′ = 0.1,when the Roe scheme is adopted, are shown in figure 5.13. For this specificproblem, the unit value for δk was not enough to eliminate completely thepresence of a small glitch at the sonic point.


(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(c) (d)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(e) (f)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

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Figure 5.12: First–order upwind density solutions for the Shock tube prob-lem: rarefaction wave with a sonic point. (a) Roe scheme, (b) Osher scheme,(c) Steger–Warming scheme, (d) Van Leer scheme, (e) Liou–Steffen (AUSM)scheme and (f) Halt–Agarwal (WPS) scheme.


(a) (b)

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Figure 5.13: First–order upwind density solutions for the Shock tube problem:rarefaction wave with a sonic point using Roe scheme. (a) Harten’s entropycorrection and (b) Wada and Liou entropy correction.

On the other hand, the entropy fix of Wada and Liou leads to the additionof diffusion at the left region to the bottom corner of the expansion fan. Thisis more easily observed in figure 5.14(a) when applied with the AUSM scheme.

The addition of diffusion in regions where it is not required must beavoided as it leads to a deterioration of the solution. In this way, if the entropycorrection of equation (5.127) is only applied if λk < δ′, no diffusion is added,apart from the region close to the sonic point. This fix is a sort of combinationof both the discussed entropy fixes. The use of this fix can be seen in figures5.14(b) and 5.15, in which the mixed entropy fix is applied with the AUSM,the Roe and the Osher scheme, and δ′ is taken equal to 0.5.

The addition of diffusion via on the entropy fix guarantees meaningfulsolutions at the expense of losing some accuracy. However, the addition ofdissipation to slowly moving waves has a positive side effect of curing thepostshock noise problem first observed by Woodward and Collella [63] andcarefully analysed by Roberts [43]. This problem occurs with nonlinear systemsof equations and is present with any scheme that recognizes the analytical


(a) (b)

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Figure 5.14: First–order upwind density solutions for the Shock tube problem:rarefaction wave with a sonic point using Liou–Steffen (AUSM) scheme. (a)Wada and Liou entropy correction and (b) mixed entropy correction.

shock jump conditions, such as the Roe and the Godunov schemes. This noiseis more accentuated for slowly moving waves and can be explained in terms ofthe discrete shock structure [43].

5.6.2 Shock tube problem: hypersonic colliding flow

The second test case consists of a Riemann problem which produces strongshock waves propagating in both directions. The initial conditions for thiscase are

UL = [ 0.1, 15√

γ, 0.1 ]T for x ≤ 0.0

UR = [ 0.1, −15√

γ, 0.1 ]T for x > 0.0

which corresponds to a strong pressure ratio (shock strength) of ≈ 379.16.This hypersonic shock tube problem (Mmax = 15.0) allows us to analyse the


(a) (b)

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1

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Figure 5.15: First–order upwind density solutions for the Shock tube problem:rarefaction wave with a sonic point using mixed entropy correction. (a) Roescheme and (b) Osher scheme.

shock–capturing capability of the schemes. The solutions, in terms of densitydistribution, are shown in figure 5.16 at time Tmax = 0.04. This correspondsto the solution after approximately 96 time–steps.

The FDS approach of, either Roe or Osher, exhibits better performancethan the FVS approach of Steger–Warming, with basically two intermediatepoints necessary to represent the shock. The Van Leer fluxes lead to a bet-ter resolution of the shock transition when compared to the Steger–Warmingscheme and similar performance as that obtained with FDS approaches. Forthis particular flow condition, the linearly degenerate wave is stationary withcontinuity of the density. The solutions computed with the FDS schemes showa small kink at the stagnation point, which corresponds to the position of thecontact “discontinuity”. It is interesting to observe that the solutions com-puted by the methods which add too much diffusion at the stationary contactdiscontinuity do not present such a kink. The Steger–Warming scheme, whichsimulates badly a contact discontinuity, as will be seen in next example, be-haves perfectly for this problem at the flat area between the two shocks. The


solution computed with the Van Leer scheme, which also is known be toodissipative, shows only a small depression there.

The AUSM and WPS, which are designed to be such that no numeri-cal dissipation is added to a stationary contact “discontinuity”, also show akink at the stagnation point. A sharp shock capture is achieved, but with avery oscillatory solution in the pre–shock regions. In fact, this overshoot at ashock is observed in other problems using AUSM or WPS. Wada and Liou [60]mention that this behavior is probably due to the fact that the mass flux inthese schemes does not directly take into account the density behind the shockwave. To correct this behaviour, they proposed some modifications, which leadto the AUSMV and the AUSMDV schemes considered above. The solutionsproduced using these schemes are showed in figure 5.17. The performance isnow similar to that of the FDS schemes, but with slightly more diffusive rep-resentation of the shock. However, the use of these schemes may lead to anexcessive numerical dissipation at any contact discontinuity and the necessityfor the additional free parameter for the AUSMDV scheme, which must beadjusted for each problem, make such schemes less attractive, in the author’sopinion. For further details of these schemes the reader should consult theWada and Liou paper [60].


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Figure 5.17: First–order upwind density solutions for the Shock tube problem:hypersonic colliding flow using Wada and Liou variants of AUSM scheme. (a)AUSMV scheme and (b) AUSMDV scheme.

5.6.3 Shock tube problem: slowly moving contact dis-continuity

The third test case considered [60] consists of a Riemann problem for whichthe initial conditions are

UL = [ 0.125, 0.3cR, 1.0 ]T for x ≤ 0.0

UR = [ 10.0, 0.3cR, 1.0 ]T for x > 0.0

This subsonic shock tube problem (Mmax = 0.30) produces a slowlymoving contact discontinuity with velocity ≈ 0.112. It is important to improvethe performance of each scheme when dealing with this linear degenerate wave.The computed solutions, in terms of density distribution, after 100 time–steps


are shown in figure (5.18). The plots for the Roe and the Osher first–orderschemes are almost identical with the contact discontinuity spreading over≈ 7 cells. The results of the AUSM and WPS schemes are very similar tothese of the FDS schemes. The results for the FVS schemes expose the wellknown defects of these approaches in dealing with a stationary or a slowlymoving contact discontinuity. A significant difference occurs at the foot of thediscontinuity, which is significantly more rounded than for the other schemes.Also, a small spurious bump at the head of the contact discontinuity is presentin the density solutions. Once more, the Van Leer scheme gives slightly betterresults than the Steger–Warming splitting.


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5.6.4 Shock tube problem: subsonic regime

In this test case, the three distinct elementary waves, namely shock, contactdiscontinuity and rarefaction waves, are present in the solution. The exampleis Sod’s [51] shock tube problem, in which the initial conditions are given by

UL = [1.0, 0.0, 1.0]T for x ≤ 0.0

UR = [0.125, 0.0, 0.1]T for x > 0.0(5.128)

A moderate shock strength of ≈ 2.031 is obtained, the speed of the con-tact discontinuity is ≈ 0.927 and the maximum Mach number is ≈ 0.93. Anindication of the performance of the various schemes when dealing with thissubsonic shock tube problem can be obtained from figure 5.19, where the massflow distribution is plotted. This variable is chosen because the profile featuresassociated with each wave are than more evident. Tmax was set to 0.20, whichis equivalent to ≈ 53 time–steps.

The hybrid FV/FD schemes present the less rounded solution at the foot ofthe expansion fan and the better shock resolution, but with a small overshootahead of the shock. The next best performance is achieved using the FDSapproaches. The use of the FV approaches, and in particular the Steger–Warming scheme, lead to the worst overall results. The constant state betweenthe expansion fan and the contact discontinuity is badly represented, with evenmore spreading of the contact discontinuity when the Steger–Warming schemeis adopted. Furthermore, the computed shock is more smeared than with theother schemes.

In order to illustrate the performance of a first–order upwind scheme forthe representation of the primitive variables: density, velocity and pressure,and the derived variables: Mach number, entropy and temperature, the solu-tions using the WPS scheme is given in figure 5.20.

The WPS was chosen as the investigation of hybrid FD and FV schemes,in the line of the AUSM and WPS, is still in its infancy. The AUSM and theWPS give almost identical solutions for this problem, with the WPS beingslightly better. These results and the results from previous applications sug-gests that AUSM and WPS might be a promising alternative to the classicalsplittings when high–resolution schemes are designed. The performance of theWPS scheme is remarkable for this problem. Indeed, the resolution of therarefaction wave and the shock wave bears almost the same accuracy as thatobtained with the use of high–resolution schemes, to be analysed in next chap-ter. However, the contact discontinuity is badly resolved and no good solutioncan be expected for inviscid flow simulations in which a contact discontinuityis presented or for any viscous flow calculations, using this first–order upwindscheme.


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Bibliography

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Chapter 6

High–ResolutionShock–Capturing Methods

6.1 Introduction

As already mentioned in chapter 2, the stability theory for linear discreteequations is normally used to check local stability of linearized equations orig-inating from truly nonlinear equations. However, in many instances, such aswhen strong discontinuities are present, instabilities can develop despite pos-sible linear stability. A stronger non–linear stability, such as total variationstability, is required to prove convergence for these cases. So far, this approachhas been well established only for scalar problems. However, despite no equiv-alent theory for general systems of equations, a broad class of schemes hasbeen successfully developed through a somewhat heuristic extension of scalarconcepts.

The initial disastrous idea of extending some standard numerical meth-ods developed under the assumption of smooth solutions led to doubts if dis-continuous profiles, i.e. very steep gradients to be resolved on an affordablediscretisation, could be accurately numerically simulated [51]. Two large fam-ilies of schemes which are conceptually different, namely shock–tracking andshock–capturing, appear which attempt to achieve high–resolution results forthe solution of hyperbolic conservation laws including discontinuities. Theshock–tracking approach employs an explicit procedure for tracking the loca-tion of discontinuities, which are treated as internal boundaries with imposedjump conditions. This represents a viable approach for one space dimension.It is possible for multi-dimension applications [14, 39], but it becomes muchmore complicated and is not sufficiently general for complex applications. Af-ter around two decades of research, the development of higher–order numericalmethods which will produce, automatically, reasonable or even good approxi-mations to discontinuities has achieved a quite mature stage. These methods

Shock–Capturing Methods 185

are called high–resolution shock–capturing methods and their success is a re-sult of the combination of mathematical and physical arguments. The high–resolution shock–capturing procedures will be employed in the remainder ofthis work.

Historically, shock–capturing methods can be divided into two categories,namely classical and modern shock–capturing methods [79]. The understand-ing of the general ideas behind both approaches, as well as a knowledge ofsome properties desirable for the numerical scheme, plays a fundamental rolein the development of these schemes. Therefore, some material covering theseaspects will be considered at the start of this chapter, before a detailed de-scription of the schemes studied in this work. The concepts will be introducedby considering a semi–discrete scheme and will be followed by some discussionabout different time integration procedures. Finally, some 1–D benchmarkproblems are analysed comparing the performance of the schemes.

It should be pointed out that, unless otherwise stated, the design princi-ples of all the schemes to be discussed refer to homogeneous hyperbolic con-servation laws and for IVPs, i.e. no source term and no reference to boundaryconditions is made. In order to keep the same high–resolution and stabilityproperties for non–homogeneous IBVPs, additional conditions must be satis-fied, which will not be discussed here. Details can be found in specific ref-erences [51, 52, 61]. Furthermore, many concepts are initially described forscalar equations, since such equations have been used as source of inspirationfor many of the existing methods and as, in most of the cases, only for suchequations are the theories well established. The extension to systems of equa-tions is normally accomplished using either the flux–difference splitting or theflux–vector splitting methods, discussed in chapter 5. The performance of theresultant schemes are evaluated by numerical experiments. Finally, it mustbe emphasized that the generic use of the term n–order scheme refers to theorder of the scheme in the smooth region of the solution. Similarly, the nota-tion upwind and symmetric high–resolution scheme, here, refers to the schemeswithout the limiter present.

6.2 Initial Considerations

Phenomena which are hyperbolic in character or are governed by pure hyper-bolic partial differential equations have little or no physical dissipation. Thesolution is characterised by the propagation of waves with little or no lossof amplitude. It is well known that any second–order, three–point numericalscheme presents unavoidable high–frequency oscillations in the neighbourhoodof discontinuities. It is also known that first–order schemes, with their inherentnumerical viscosity, damp the high–frequency components of the solution and,


in doing so, they smooth out strong gradients. This behavior can be observedin figure 6.1 where the solution of the IVP given by the Burgers equation anda sinusoidal initial data is shown.

(a) (b)

u

x

u

x

Figure 6.1: Typical numerical solutions of 1–D Burgers equation using classicalfinite difference schemes. (a) First–order upwind scheme; (b) Lax–Wendroffscheme.

The solution at two distinct time levels is shown, one while the solution isstill smooth and another after it has developed into a shock. The performanceof a typical first–order scheme is seen in figure 6.1(a), where the predominantdiffusive nature of the approximation has the effect of smearing the solutionas time evolves. On the other hand, a second–order solution, figure 6.1(b), ischaracterised by the presence of oscillations close to the shock, as a result ofthe dominant dispersive nature of the error in the approximation.

One approach for analysing the behavior of a numerical scheme consistsin utilizing the method of the equivalent differential equation [22, 76]. Byanalysing the first term of the truncation error, one can easily recognize thedominant dissipation or dispersion nature of the scheme. Another very simpleand illustrative approach, when dealing with scalar hyperbolic equations, refersto the interpretation of the discrete scheme as a combination of shock trackingand polynomial interpolation, see appendix C.

Some of the most significant contributions, which set the theoretical basis,for the design of high–resolution schemes can be found in Courant–Friedrichs–Lewy [9]; Courant–Isaacson–Rees [10], with the introduction of the CFL stabil-ity condition and of the scheme which originates the family of upwind schemes;Von Neumann and Richtmyer [72]; Lax [30]; Lax–Wendroff [32], concerningthe concept of artificial viscosity, combined space–time discretisation, stabi-lization of second–order centered schemes, discrete conservation and manyother fundamental ideas; Godunov [15]; Lax [31]; Van Leer [66, 67, 68, 70],


regarding the solution of the problem as an interaction between discretisationmethods and successive local Riemann solutions, monotonicity, entropy con-dition and second–order Godunov–type schemes; Beam–Warming [3, 4]; Ste-ger [56]; Jameson–Schmidt–Turkel [28], regarding to second–order separatedspace–time discretisation, addition of non–linear explicit artificial dissipationand its extension to multi dimensions using generic discretisation; Boris–Book[5]; Van Leer [66]; Harten–Hyman–Lax [20]; Harten [17, 18], regarding theintroduction of non–linear limiters and the concept of total variation dimin-ishing (TVD) schemes; Steger–Warming [57]; Roe [48]; Engquist–Osher [13];Osher [40] who proposed different approaches to extend upwind concepts tonon–linear scalar and systems of equations, through the concepts of flux vectorsplitting and approximate Riemann solvers. Although, there are many omis-sions in the coverage given above, most of the permanent material, concerningto the theoretical development of high–resolution shock–capturing schemescan be found in the given references and those mentioned in referenced pa-pers. Some of these theoretical concepts have already been briefly introducedin chapter 5, and some other concepts will be introduced below.

6.2.1 Classical shock–capturing schemes

Although monotone schemes possess many desirable properties for the calcu-lation of discontinuous solutions, they are only first–order accurate [23], as willbe discussed later in this chapter. The diffusive nature of first–order schemesmean that they cannot be used to produce accurate results with an affordablemesh spacing for real problems. On the other hand, the use of higher–orderschemes lead to spurious oscillations in the results. The idea then is to pro-duce high–resolution methods, via an attempt to modify high–order methodsto cure their deficiencies. In order to do so, Von Neumann and Richtmyer[72] introduced the concept of artificial viscosity. This was also suggested byLax and Wendroff [32], who further developed this concept and showed theimportance of considering the scheme in discrete conservation form.

The idea of introducing an artificial dissipation, which resembles the phys-ical viscosity, represents a conceptual breakthrough towards the developmentof high–resolution schemes. In the schemes referred to as classical, such numer-ical dissipation terms are either linear, so that the same amount of diffusion isapplied at all grid points, or contain empirically adjustable parameters [79]. Asa result, the classical shock–capturing schemes can only successfully addressproblems with smooth or weak shock solutions. They are not robust enoughto deal with strong discontinuities, resulting in oscillations near such featuresand/or nonlinear instabilities [79]. Finally, they are problem–dependent whichdisqualifies their use for applications for which the analyst has no previousknowledge.


6.2.2 Advanced shock–capturing schemes

The key difference between the classical and the modern shock–capturingschemes is the utilization by the latter of more elaborate non–linear devices,which allow an adaptive control of the numerical dissipation added to stabilizeand eliminate oscillations in the solution.

Advanced shock–capturing methods can be generically classified as artifi-cial viscosity type schemes, algebraic schemes or geometric schemes. The firsttype of scheme still relies on the explicit addition of an artificial dissipation,but now consisting of an ingenious blending of fourth– and second–order termswhich are activated non–linearly by the use of a switch [25, 28], normally basedon pressure. These schemes have been extensively applied with success, mainlyfor transonic simulations. Recent improvements, initially suggested by Turkel[64], expanded its success for supersonic and hypersonic applications [46, 59].

The division into algebraic and geometric approaches is based on themethod used to compute the interface fluxes and was originally suggested byGoodman and Le Veque [16]. In both approaches, the incorporation of someideas drawn from the theory of characteristics plays a major role in enablingthe development of most of the modern shock–capturing methods. There isa wide range of such methods, which are often very closely connected. Themain difference from the artificial viscosity approach is the implicit inclusionof the numerical diffusion in the discrete equation, eliminating or reducing thenumber of free parameters which need to be tuned.

A review of many high–resolution methods is presented in this chapter.For a more comprehensive review, the work of Woodward and Colella is recom-mended [75], where a detailed analysis of a few schemes is presented for both1–D and 2–D simulations; Zalesak [81] and Yang and Przekwas [77], where alarge number of schemes are compared when applied to solve linear advectionand Burgers equations, respectively. Also, the comprehensive work of Yee [79]is highly recommended. This presents a large class of high–resolution explicitand implicit shock–capturing methods with applications to steady–state andtransient, perfect gases, equilibrium real gases and nonequilibrium flows.

6.2.3 Non–oscillatory properties: background

Before tackling the broad subject of constructing high–order shock–capturingmethods, some very useful and frequently required concepts, must be intro-duced. These concepts are indeed properties of the true solution to scalarconservation laws, which means that it is reasonable to impose them on thenumerical solution as well. As already mentioned, a large number of effec-tive schemes have been developed with the use of these properties. For morecomprehensive definitions, see for instance references [23, 33, 55].


Consider a one–dimensional scalar conservation law

∂u

∂t+

∂F (u)

∂x= 0 (6.1)

A numerical method applied to (6.1) can be expressed in the general form

un+1I = H(un

k ; k) (6.2)

where k represents the discrete points and H a discrete operator.

Monotone methods

The scheme (6.2) is called monotone nondecreasing if

∂H(unk ; k)

∂unI

≥ 0 ∀ I, k, un (6.3)

A similar definition holds for monotone nonincreasing schemes. This meansthat the function H is such that

If vnI ≥ un

I ∀ IH

=⇒ vn+1I ≥ un+1

I ∀ I (6.4)

Harten et al [20] show that a converged solution to a consistent monotonemethod satisfies the entropy condition defined in section 5.2.4. Such a resultwhen incorporated in Lax–Wendroff theorem, section 5.3.2, defines that:

“If the numerical solution u(x, t) computed with a monotone con-sistent and conservative method converges, boundedly almost ev-erywhere, to a function v(x, t) as ∆x and ∆t approach zero, thenv(x, t) is a physically acceptable weak solution of the conservationlaw.”

Thus, monotone methods possess many desirable properties for the calculationof discontinuous solutions. However, this class of methods is too restrictive asa theorem, due to Godunov [15] states that:

“A monotone method in conservation form is at most first–orderaccurate.”

This theorem represents a major restriction in the design of numerical methodsfor the solution of hyperbolic equations.


Monotonicity preserving methods

One way to get around the limitation imposed by the previous theorem con-sists in considering a weaker requirement of a numerical method, such as theintroduction of the monotonicity preserving concept. The numerical methodgiven in (6.2) is monotonicity preserving, either nondecreasing or nonincreas-ing, if for any monotone data u0, the solution un is also monotone for all n. Itmeans that

If u0I ≥ u0

I+1 ∀ IH(n)

=⇒ unI ≥ un

I+1 ∀ I, n (6.5)

and in particular, that no oscillations arise near an isolated propagating dis-continuity, since Riemann’s problem initial data is monotone. Except for thelinear constant coefficient case, in which monotonicity is equivalent to mono-tonicity preserving [15], Van Leer [67] shows that it is possible to constructmonotonicity preserving schemes, by introducing non–linear terms, which haveorder higher than one. This attractive property opens the possibility for thedesign of the so called high–resolution methods.

Total variation diminishing (TVD) methods

Another extremely important concept is based on the mathematical property[31] that the total variation, TV, of any physical admissible solution of a scalarconservation law,

TV =∫ ∞

−∞|∂u

∂x| dx (6.6)

can never increase. Correspondingly, the numerical method (6.2) is called totalvariation diminishing, or more correctly total variation nonincreasing if

TV(un+1) ≤ TV(un) ∀n (6.7)

with

TV(u) =∑

I

|uI+1 − uI | (6.8)

Harten in [17, 18] developed some mathematical conditions which enabledthe proof of the TVD property of a three–point numerical scheme. These con-ditions were further generalised for multi–point schemes by Jameson and Lax[26]. Harten also proved that any TVD scheme is monotonicity preserving and


so does not allow the appearance of spurious oscillations near single disconti-nuities. Consider a semi–discrete scheme cast in the form

∆xduI

dt= BI−1 I(uI−1 − uI) + BI+1 I(uI+1 − uI) (6.9)

The sufficient condition, which (6.9) has to satisfy in order that (6.7) holdstrue, is that BI−1 I ≥ 0 and BI+1 I ≥ 0, [18, 60].

It can be proved that any three–point TVD scheme is at most first–orderaccurate and that the Roe’s first–order upwind scheme is the least diffusivefirst–order TVD scheme [23, 24], see section 6.4.1. In this way, if a schemeis written in the form given in (6.9), the coefficients BI−1 I and BI+1 I mustdepend on more than two points in order to obtain a higher–order TVD scheme.This leads to the definition of limiters, to be discussed in a later section.

Local extremum diminishing (LED) methods

Another concept, closely connected with TVD in one–dimension, but which canbe readily generalized to multi–dimensions, is that of local extremum dimin-ishing, LED, methods [24, 26]. Suppose a numerical scheme can be expressedas

∆xduI

dt=∑

k

BkI(uk − uI) (6.10)

where BkI is a coefficient with dependence on a compact stencil of points andsuch that

BkI ≥ 0 ∀ k (6.11)

This condition is sufficient to guarantee that a local maximum (minimum)cannot increase (decrease), and so the scheme is LED. For instance, if a localmaximum exists at point I, i.e. uI ≥ uk ∀ k, then (uk − uI) ≤ 0 and from(6.10), (6.11) it can be shown that duI/dt ≤ 0. Note that if uI ≥ 0 ∀I, theglobal minimum is positive, and, as it cannot decrease, the positivity of thesolution is ensured.

Laney and Caughey [29] observed that each extremum appears in thevariation of the two segments which contain such extrema, and if the endvalues of each of these segments are fixed, the total variation (6.8) can beexpressed [24] according to

TV (u) = 2(∑

maxima −∑

minima) (6.12)


It is readily observed that if a one–dimensional scheme is LED, it is also TVD.However, it is not simple to verify the condition (6.7) for multidimensionalschemes, while condition (6.11) can still be demonstrated. Since LED guaran-tees TVD, the notation LED or TVD will be used without distinguishing inthis chapter for 1–D scalar schemes.

Summary

Osher and Chakravarthy [41], Jameson and Lax [26] prove that TVD meth-ods, no matter how they are constructed, must in fact degenerate to first–orderaccuracy at extreme points. Two other concepts used to design higher–orderschemes without such limitation, and which will not be studied here, are nowmentioned for completeness. The total variation bounded, TVB, schemes ofShu [52] and the essentially nonoscillatory, ENO, schemes of Harten et al[19, 21, 54]. The TVB condition requires that the total variation should bebounded by a fixed constant, depending on the initial data, at any time. Thisrequirement is less restrict than the TVD criterion and allows TVB schemesto be uniformly higher–order accurate in space, including extrema points.The ENO concept represents an extension of the Godunov approach basedon higher–order reconstruction of the solution, in a similar way to MUSCL[70] schemes to be presented in section 6.5.1 and PPM [8], but with the uniquefeature of using an adaptive stencil interpolation to obtain information auto-matically from regions of smoothness when the solution develops discontinu-ities. Like TVB schemes, these essentially nonoscillatory schemes can retainthe spatial accuracy even at points of extrema, by allowing occasionally accen-tuation of local extrema.

The additional operation count involved in the computation using ENOschemes and the promise of extra difficulties and complexities which need tobe faced extending ENO schemes for implementation on multidimensional un-structured grids, make it less attractive in the present work. Furthermore,these two approaches, TVB and ENO, are still in the development stage,while TVD is well established in the sense that different formulations havebeen applied to a wide range of CFD problems in multi–dimensions [79].

To sum up the concepts discussed, it is illustrative to present the hier-archy of conservative schemes for hyperbolic conservation laws given by Yee’sdiagram [79] and shown in figure 6.2.

In figure 6.2, ST represents the set of all existing conservative schemes ofany order for hyperbolic conservation laws. This set can be broken into thesubset SUP of all upwind–based schemes of any order and the subset SC ofcentered–based schemes. The other subsets are: SM , the set of all monotoneschemes; STV D, the set of all TVD schemes; STV B, the set of all TVB schemes;SENO, the set of all ENO schemes.


SENOSTVB

SC SUP

SM

STVD

TS

STVB SENO TSSTVDSM

U U U U

TS SC SUP= U

Figure 6.2: Hierarchy of the conservative schemes for hyperbolic conservationlaws.

It must be noted that none of the previous properties, with the excep-tion of monotonicity, ensures that an entropy condition is satisfied. However,a converged numerical solution obtained with a conservative high–resolutionscheme is expected to be physically correct provided that the original first–order scheme used in its construction satisfies the entropy condition [60]. Also,a general trend, which is encountered when schemes moves from monotone to-wards ENO, is that the higher–order is achieved at the expense of a biggerstencil of points, more complexity and less flexibility of the computational im-plementation. These features connected with memory and CPU requirementsmust be carefully analysed to provide a fair comparison of the different classesof methods.

6.3 Switched Artificial Viscosity Approaches

The concept of artificial viscosity, already mentioned in section 6.2.2, plays anessential role in attempting to stabilize and/or eliminate spurious oscillationsat discontinuities when space–centered discretisations are adopted.

A semi–discrete numerical scheme for the solution of a system of conser-vation laws can in general be written in conservative form, see section 5.3.2,according to

∆xdUI

dt= −

FFF I+1/2 − FFF I−1/2

(6.13)


with the numerical flux function expressed as the sum of a central differencediscretisation of the physical flux function plus a dissipation term, i.e.

FFF I+1/2 =1

2

(F I + F I+1) − FFF I+1/2

(6.14)

For the three–point central difference method FFF I+1/2 is identically zeroand if an explicit Euler time integration is adopted the resultant scheme isunstable [22]. Some other possibilities for the time integration, such as Runge–Kutta multistage schemes, are normally used with central differences in orderto enhance the stability range for (6.13), and will be investigated later in section6.7. A detailed analysis of such schemes shows that high frequency componentsof the solution remain undamped, and as the scheme is second–order in spacethe adopted cure must be the addition of a higher–order dissipation term sothat the basic accuracy is not degraded.

6.3.1 Scalar equation

A scalar conservation law is considered initially. In this case, a practical formu-lation for smooth flows can be produced by computing the artificial dissipativeterm, [23], as

F (CD)I+1/2 = αI+1/2ǫ

(4)I+1/2

[∆uI+1/2 − (1/2)(∆uI+3/2 + ∆uI−1/2)

](6.15)

where αI+1/2 is a scaling factor to be defined later and ǫ(4) is a free parameterspecified by the analyst. The artificial dissipative term is obtained by sub-tracting an average of neighbouring differences from ∆uI+1/2. An alternativeformulation is obtained by taking the one–step Lax–Wendroff flux [32] insteadof the central difference flux. This corresponds to the numerical flux (6.14),with the dissipative term

F (LW )I+1/2 =

∆t

∆xa2

I+1/2∆uI+1/2 (6.16)

where aI+1/2 denotes the Jacobian of the convective flux at the cell interfaceI + 1/2. The dissipative term arises from a Taylor expansion in time, up tosecond–order, following some algebraic manipulation. It has been designed inan attempt to minimize the truncation error of a three–point scheme.

The dissipative term in (6.15) and (6.16) suppresses the even–odd modesthat appear with central differences and stabilizes the scheme allowing it toreach a steady–state. Furthermore, the diffusion terms also prevent the ap-pearance of non–physical solutions. Nevertheless, the selective addition of


further dissipation is required to prevent the appearance of oscillations whenthe algorithm is used to simulate flows involving discontinuous solutions.

As with the first–order upwind scheme, equation (5.55), the dissipativeterm of a generic diffusive scheme, at most first–order accurate, can be writtenas

F (1)I+1/2 = αI+1/2(uI+1 − uI) (6.17)

where αI+1/2 can be considered as the numerical viscosity coefficient of thescheme, and defines uniquely the three–point scheme. These schemes, suchas the first–order upwind scheme, give non–oscillatory shock transition but atan expense of being too diffusive. MacCormack and Baldwin [38] suggest totake αI+1/2 proportional to a second derivative of the flow variable in order tocontrol the amount of diffusion.

Inspired by the application of the linear fourth–order dissipation given in(6.15), which was introduced by Steger [56], and by the non–linear artificialviscosity applied by MacCormack and Baldwin [38], Jameson et al [28] builtan adaptive artificial dissipation that combines both dissipative terms. Con-sidering some slight modifications proposed by Swanson and Turkel [59] andby Peraire et al [46], the artificial dissipative term can then be written as

F (CD)I+1/2 = αI+1/2

ǫ(2)I+1/2∆uI+1/2 + ǫ

(4)I+1/2

[∆uI+1/2

−(1/2)(∆uI+3/2 + ∆uI−1/2)] (6.18)

The parameters ǫ(2)I+1/2, ǫ

(4)I+1/2 are adapted to the flow and they are defined

according to

ǫ(2)I+1/2 = µ(2)max(ΥI , ΥI+1) ǫ

(4)I+1/2 = max[0, (µ(4) − ǫ

(2)I+1/2)] (6.19)

with

ΥI =|uI+1 − 2uI + uI−1|

(1 − θ)(|uI+1 − uI | + |uI − uI−1|) + θ(uI+1 + 2uI + uI−1) + ζ(6.20)

In the above equations, µ(2) and µ(4) are user specified coefficients. The factorΥI is a sensor designed to detect discontinuities. The parameter θ represents aweighting coefficient, 0 ≤ θ ≤ 1, and ζ is a small constant in order to avoid theappearance of a zero in the denominator. For the scalar equation, a natural


choice for the diffusive parameter is αI+1/2 = |aI+1/2|, i.e. to take (6.17) as thefirst–order upwind scheme (5.55).

Equation (6.20) has been built in such way that ΥI should only be signif-icant in regions of strong gradients, controlling the amount of artificial dissi-pation to be added and preserving the second–order of accuracy of the schemein smooth regions. In the presence of these strong gradients, the second–orderdifference factor ǫ

(2)I+1/2 will be activated and the fourth–order dissipation will

be turned off, i.e. ǫ(4)I+1/2 ≡ 0. This is important as only the second–order term

is desired near the strong gradients and the fourth–order term must be avoidedas it produces oscillations there.

The purpose of the, basically linear, fourth–order difference dissipationactivated throughout the smooth part of the solution is to prevent nonlinearinstabilities by damping high frequencies modes and so allowing the schemeto approach a steady–state. For transient flows, one can set this dissipationidentically zero [59]. On the other hand, the second–order difference dissipationterm is non–linear and designed in order to introduce an entropy–like conditionand avoiding the appearance of oscillations in the vicinity of strong gradients inthe flow. When the Lax–Wendroff scheme is used, the second–order viscosityterm is also necessary, so that shocks can be captured without oscillations.The numerical dissipation flux for the fully discretised explicit Lax–Wendroffscheme is then given by

F (LW )I+1/2 =

∆t

∆xa2

I+1/2∆uI+1/2 + αI+1/2ǫ(2)I+1/2∆uI+1/2 (6.21)

6.3.2 System of Equations

All the expressions presented for the scalar case can be directly extended to dealwith systems of equations, with the switch (6.20) being in general computedin terms of the pressure field. However, the choice of the scaling parameterαI+1/2 is not as natural as in the scalar counterpart and some possible choicesare now discussed.

Scalar artificial viscosity

The simplest choice for the scaling factor consists in taking the spectral radius,or maximum eigenvalue λmax, of the Jacobian matrix A of the convective flux,computed here as

(λmax)I+1/2 =1

2[(|uI | + cI) + (|uI+1| + cI+1)] (6.22)


with uI , cI denoting the values of the fluid velocity and speed of sound at nodeI, respectively. The corresponding dissipative scheme, (6.17), used to buildthe final scheme (6.18) or (6.21) has a diffusive term given by

FFF (1)

I+1/2 = (λmax)I+1/2(UI+1 − UI) (6.23)

This choice, with a scalar coefficient, does not distinguish the different fields ofthe system, adding the same amount of diffusion to all of them. It representsa very cheap option and was the original one used by Jameson et al [25, 28],and has been used since then by many CFD practitioners.

Matrix artificial viscosity

Through a careful examination on the connection between upwind schemesand central difference plus artificial viscosity schemes, Turkel [64], Swansonand Turkel [59] proposed a modification on the above scheme, which mimicsthe upwind scheme in the vicinity of discontinuities. As observed by Turkel[64], the adoption of a scalar coefficient, based on the spectral radius of theJacobian matrix, when defining the scaling factor α adds too much viscosityto the slower waves. He suggested then to compute α as a matrix coefficient,i.e.

αI+1/2 = |A| = R|Λ|R−1 (6.24)

where |A| · ∆U can be computed as described in the appendix B. A cutoff,or entropy fix, is necessary to prevent difficulties near stagnation and sonicpoints. The procedure given in equation (5.85) is adopted here. The use of(6.24) is equivalent to taking Roe’s first–order upwind scheme (5.82) as the

dissipative scheme FFF (1)

I+1/2. This close connection to upwind–based schemeswas used by Swanson and Turkel [59] to show certain conditions, for the choice

of ǫ(2)I+1/2 and ǫ

(4)I+1/2, that can ensure TVD of this form of artificial viscosity.

The adoption of the matrix artificial dissipation has been found, [45, 59], to beimportant in enhancing solution accuracy for viscous simulations, but with justminor improvement for the inviscid case. However, it represents an expensivealternative, with the switched artificial viscosity approach losing partially itsmain advantage of being cost effective.

Turkel [64] pointed out that the use of a matrix coefficient scaling factordoes not allow for a constant enthalpy solution and so enthalpy damping cannotbe used. Other possibilities for the switch (6.20) can be considered by usingother variable such as temperature, which also detects contact discontinuities,or even using different switches for linear and non–linear waves, when usingthe matrix approach [59], but these are not attempted here.


Flux–splitting artificial viscosity

One alternative way to introduce artificial viscosity has been recently proposedby Jameson [24], who coined the term CUSP, which means convective upwindand split pressure. This scheme was once more inspired by upwind schemesand involves the construction of an artificial dissipation which resembles thatfound in the AUSM flux splitting discussed in section 5.4.3. Following thenomenclature set in section 5.4.3, the diffusive term of the dissipative scheme,(6.17), can be computed using

FFF (1)

I+1/2 = ǫ(2)I+1/2 [ f1(M)cI+1/2(ΘI+1 − ΘI) + f2(M)(PI+1 − PI)∆ ] (6.25)

where α ≡ 1 in equations (6.18) or (6.21) for the second–order differencediffusion, since the scaling is intrinsically incorporated in the functions f1, f2.The interface Mach number M is computed as

M = MI+1/2 = (uI)−/cI + (uI+1)

+/cI+1 (6.26)

with (uk)± determined according to (5.105). The interface speed of sound is

obtained using [44]

cI+1/2 =1

2cI [1 − f2(M)] + cI+1 [1 + f2(M)] (6.27)

The blending functions f1 and f2 are defined in order to ensure a suitabledomain of dependence for subsonic flows and full upwinding for supersonicregimes. They are computed as

f1(M) =

a0 + a2M2+ a4M

4 |M | < 1

|M | |M | ≥ 1(6.28)

f2(M) =

M(3 − M

2)

2|M | < 1

sign(M) |M | ≥ 1

(6.29)

where

a2 =3

2− 2a0 a4 = a0 −

1

2(6.30)


with a0 controlling the diffusion at M = 0. Jameson [24] suggests the choicea0 = 1/4 for transonic simulations and a0 = 1/2 for high speed flow simu-lations. The fourth–order artificial dissipative term (6.15) or the dissipativeterm of the Lax–Wendroff scheme (6.16) is added in a similar way to thatdescribed for the scalar or matrix artificial viscosity schemes.

The diffusion corresponding to the convective term Θ is similar to thatpresent for the scalar artificial viscosity scheme, with a modification to thescaling. The separate treatment of the pressure term represents the additionalmodification necessary to produce perfect upwinding in the supersonic regime.

The cost involved in this scheme is intermediate when compared to thesimple scalar or the more elaborate matrix coefficients. Jameson [24] and Tat-sumi et al [62] found excellent shock–capturing properties of the scheme, withgood oscillation control and sharp shock resolution, with the scheme perform-ing even better for hypersonic simulations. They also found an excellent con-vergence rate towards a steady–state and they pointed out that this schemeretains the property of the original scalar scheme of being compatible withconstant stagnation enthalpy at the steady–state.

Of course following this approach, any upwind scheme defined in chapter5 can be used as the diffusive scheme. The scaling factor for the fourth–order difference term can be taken as the simple scalar coefficient with littleeffect on shock resolution [64]. The centered–based schemes with the additionof a switched artificial viscosity, using any of the options discussed or theirvariants, has been applied with success in the whole range of flow regimes.But there remains the necessity for tuning free parameters which represents,in the author’s opinion, a drawback of such approaches, as the scheme loses thedesirable property of robustness. Other authors do not share the same opinionand see the external adjustable parameters as one more flexible element of thescheme, which allows the analyst to control the amount of dissipation added.Of course, this argument is fallacious since free parameters can be introducedin any scheme and cannot be seen as a privilege of a specific approach.

6.4 Flux–Limited Methods or Algebraic Ap-

proaches

In the flux–limited approach, also referred to as a hybrid scheme [75], a higher–order scheme, which is very accurate in smooth regions of the flow but badlybehaved at discontinuities, is blended with a lower–order scheme, which hasapproximately opposite behavior. Accordingly, the resultant numerical fluxcan be written in the form


FFF I+1/2 = FFF (L)I+1/2 + LI+1/2

AAAI+1/2︷︸︸︷( FFF (H)

I+1/2 − FFF (L)I+1/2)︸︷︷︸

FFF (C)

I+1/2

(6.31)

where FFF (L)I+1/2 and FFF (H)

I+1/2 refer to a lower– and higher–order schemes. Thekey point of such procedures is the guarantee of the monotonicity propertyof the lower–order scheme FFF (L)

I+1/2 and the definition of the correction factorL. This factor is designed to limit the anti–diffusive flux AAAI+1/2 leading to

the limited corrective flux FFF (C)I+1/2, which gives rise to the name flux–limited

methods. Generally speaking, if the solution is smooth, then L should be nearone, while close to discontinuities it must be near zero. Note that equation(6.31) can be rewritten as

FFF I+1/2 = FFF (H)I+1/2 + (I − LI+1/2)( FFF (L)

I+1/2 − FFF (H)I+1/2) (6.32)

where I is a unitary vector. Thus, the scheme given in (6.32) can be inter-preted as the addition of a limited amount of diffusion, which is negligiblein smooth flow and large near discontinuities. Although, this resembles theswitched artificial viscosity approach, the added term is not motivated by theanalogy with a more realistic physical model for the flow than that given bythe inviscid equation. Instead, this term is designed specifically in order togive the sharpest possible monotone discontinuity profile.

There are two general flux–limited techniques: the predictor–correctorFCT, flux corrector transport, method of Boris and Book [5] and later gen-eralized for multi–dimension by Zalesak [80], and the one–step technique firstproposed by Van Leer [66, 67], later set into a more mathematical frameworkby Harten [17, 18] with the concept of total variation diminishing and fur-ther developed by Roe [50], Davis [11], Sweby [60], Jameson and Lax [26],Chakravarthy and Osher [7], Yee [78] and many others.

The FCT approach is less firmly grounded from a theoretical point of view,however its flexibility and the generality allowed Zalesak [80] to extend theBoris–Book [5] scheme to multi–dimensions and Lohner et al [35] to incorporateFCT into an unstructured grid finite element code. The FCT approach willnot be studied here and the previous referenced literature should be read forfurther details on FCT implementation and performance.

The major difference between these two flux–limited procedures are con-cerned with the fact that the FCT scheme uses a lower–order predicted solutionin order to define the limited anti–diffusive flux, while the family of TVD–likeschemes uses the old time–level solution. This is the reason for the two–stepsrequired by FCT schemes. Another difference concerns the fact that, in the


FCT algorithm, the limiter, or corrective factor, L is constrained by a unityvalue, while in the one–step approach a wider range of values is normallyallowed for L. This can lead to better results, as will be seen later.

Although, the TVD concept has been widely used in the design of one–stepflux–limited schemes, it does not readily generalise to multi-dimensions, as al-ready mentioned. On the other hand, the LED concept, which assures TVD inone–dimension, is directly extendible to multi–dimensions for general meshes.Whenever possible, I will attempt to follow the framework of Jameson’s LED[24] procedure for the design of high–resolution schemes.

In what follows, high–resolution LED schemes are derived in much thesame way as in the FCT approach, i.e. the application of a lower–order schemesupplemented by the addition of a limited anti–diffusive flux, (6.31). This pro-cedure was adopted by Sweby [60], who also presented a detailed discussion onthe necessary and sufficient conditions required for the limiter functions suchthat the final high–order scheme satisfies TVD sufficient conditions. In thisway, Sweby unified several independently proposed high–order scalar TVDschemes, which use Roe’s flux difference splitting approach, to extend thescheme to handle systems of equations. Liou [34] did similar work, but he con-centrated on procedures which are extended for systems of equations throughthe use of flux vector splitting approaches. In this way, any first–order upwindscheme which can be cast in the general FVS form given by (5.91), includingthe Liou and Steffen splitting AUSM when written as in equation (5.101), canbe extended to a higher–order TVD scheme using Liou’s alternative approach.Full details are given in the reference mentioned above.

6.4.1 Conditions for low–order LED schemes

Consider the semi–discrete approximation to the 1–D scalar conservation law(6.1), in conservative form

∆xduI

dt= −

FE

I+1/2 − FEI−1/2

n(6.33)

where FEI+1/2 = FE

I+1/2(uI , uI+1) denotes the numerical flux of a three–pointdiscrete entropy scheme, or E–scheme, [41, 60], i.e. a numerical scheme thatcan be proved to accept only physically correct discontinuities. These schemesmay be characterized by the inequality [23, 60],

sign(uI+1 − uI)[FEI+1/2 − F (u)] ≤ 0 ∀ u | uI ≤ u ≤ uI+1 (6.34)

Using similar definitions for the flux differences as presented in section 5.4.1,it is possible to write


∆F−I+1/2 = +(FE

I+1/2 − FI) = α−I+1/2∆uI+1/2

∆F+I−1/2 = −(FE

I−1/2 − FI) = α+I−1/2∆uI−1/2

(6.35)

Using the inequality (6.34), it can be observed that the local wave speedsare such that: α−

I+1/2 ≤ 0 and α+I−1/2 ≥ 0, justifying the signs −, + as the

superscript. Making use of (6.35), expression (6.33) can be written in thegeneral form (6.10), i.e.

∆xduI

dt= (+α+

I−1/2)︸︷︷︸BI−1 I

(uI−1 − uI)︸︷︷︸−∆uI−1/2

+ (−α−I+1/2)︸︷︷︸

BI+1 I

(uI+1 − uI)︸︷︷︸∆uI+1/2

(6.36)

where BI−1 I ≥ 0, BI+1 I ≥ 0. This is equivalent to (6.11), showing that anyE–scheme is LED, and so TVD.

If the wave speed a(u) = ∂F/∂u is approximated according to

aI+1/2 =

∆FI+1/2/∆uI+1/2 if uI+1 6= uI

∂F/∂u|I if uI+1 = uI

(6.37)

then the numerical flux, for a generic diffusive scheme, can be written as

FI+1/2 =1

2

(2FI + aI+1/2∆uI+1/2) − αI+1/2∆uI+1/2

(6.38)

and the corresponding semi–discrete scheme, (6.33), is

∆xduI

dt=(

αI+1/2 − aI+1/2

2

)∆uI+1/2 −

(αI−1/2 + aI−1/2

2

)∆uI−1/2 (6.39)

The LED condition (6.11) requires that the terms inside the brackets must benon–negative, which is equivalent to requiring

αI+1/2 ≥ |aI+1/2| (6.40)

Thus Roe’s first–order upwind scheme, αI+1/2 = |aI+1/2|, is the least diffusivescheme that satisfies the LED property. However, it is not an E–scheme, asthe condition (6.34) does not hold when the interval (I, I + 1) contains asonic point, [23]. An entropy fix, as mentioned in section 5.4.1 must be addedto make it an entropy scheme. It can be proved that three–point monotoneschemes belong to the class of entropy schemes [23]. In this sense the adoptionof a monotonic upwinding scheme as a lower–order scheme, is a natural choicewhen designing non–oscillatory high–resolution schemes.


6.4.2 Conditions for high–order LED schemes

As already discussed in section 6.3, a higher–order numerical scheme can bedesigned using the artificial dissipation given in (6.15), and can be written as

F (H)I+1/2 =

1

2

(2FI + aI+1/2∆uI+1/2)

−αI+1/2

[∆uI+1/2 − (1/2)(∆uI+3/2 + ∆uI−1/2)

] (6.41)

where no free parameter is considered here and ǫ(4)I+1/2 ≡ 1. The resultant

scheme using (6.41) does not satisfy LED sufficient conditions. Assuming thegeneral lower–order scheme with the numerical flux given in equation (6.38)and the higher–order scheme (6.41), a limited higher–order scheme, accordingto (6.31), has a numerical flux which can be written as

F (H)I+1/2 =

1

2

(2FI + aI+1/2∆uI+1/2)

−αI+1/2

[∆uI+1/2 − L(2)(∆uI+3/2, ∆uI−1/2)

] (6.42)

where L(2)(a, b) is no longer the simple arithmetic average that appears inequation (6.41), but a limited average of a and b. This limited average satisfythe natural properties of an average:

(P1) L(2)(a, b) = L(2)(b, a);

(P2) L(2)(αa, αb) = αL(2)(a, b);

(P3) L(2)(a, a) = a;

(6.43)

which can be generalised directly for a function having n parameters as argu-ments, for instance the three–parameter function L(3)(a, b, c). It is convenientto introduce the following notation

Φ(1)(r) = L(2)(1, r) Φ(2)(r−, r+) = L(3)(r

−, 1, r+) (6.44)

and from property P.2, it follows that

Φ(1)(r) = r Φ(1)

(1

r

)(6.45)


Here Φ(1) is a one–parameter function, r denotes a centered ratio of differencesand r± are the ratio of forward and backward differences defined as

r+I+1/2 =

∆uI+3/2

∆uI+1/2

; r−I+1/2 =∆uI−1/2

∆uI+1/2

;

rI+1/2 =r+I+1/2

r−I+1/2

=∆uI+3/2

∆uI−1/2

(6.46)

Making use of these notations, the semi–discrete scalar scheme adopting(6.42) reduces to

∆xduI

dt=

1

2

[αI+1/2 − aI+1/2 + αI−1/2Φ(1)

(1

rI−1/2

)]∆uI+1/2

−[αI−1/2 + aI−1/2 + αI+1/2Φ(1)

(rI+1/2

)]∆uI−1/2

(6.47)

The scheme given in (6.47) is LED if the terms inside the square brackets arepositive, which is true whenever

αI+1/2 ≥ |aI+1/2| ∀ I and Φ(r), Φ(1

r) ≥ 0 ∀ r (6.48)

The first condition above is assured by the choice of an LED lower–orderscheme, (6.40). The second condition in (6.48) requires the definition of anextra property for the limited average L(2), i.e.

(P4) L(2)(a, b) = 0 ∀ a, b with opposite signs. (6.49)

It is important to observe that the positivity of the terms inside the squarebrackets in (6.47) is reinforced by the presence of the term involving the limiterΦ, as a result of the second condition given in (6.48).

In fact, a more natural numerical flux for the non–linear scalar problem,would consider

F (H)I+1/2 =

1

2

(2FI + aI+1/2∆uI+1/2) −

[αI+1/2∆uI+1/2

−L(2)(αI+3/2∆uI+3/2, αI−1/2∆uI−1/2)] (6.50)


instead of (6.42), with L(2) imposing constraints directly on the flux variations.However, it is directly verifiable that previous arguments remain valid. Fur-thermore, numerical experiments presented by Davis [11] and Yee [78] with2–D Euler equations, indicate a better behavior of the schemes which use thelimited average over the variations of the dependent variables, and (6.42) isthus preferable.

The high–order scheme (6.41) might have been devised with the dissi-pation term obtained by subtracting an average of the differences ∆uI−1/2,∆uI+1/2, ∆uI+3/2 from ∆uI+1/2. This would lead to a limited higher–ordernumerical flux

F (H)I+1/2 =

1

2

(2FI + aI+1/2∆uI+1/2) − αI+1/2

[∆uI+1/2

−L(3)(∆uI+3/2, ∆uI+1/2, ∆uI−1/2)] (6.51)

where a three–parameter limited average L(3) is used in order to prevent os-cillations to get an LED scheme. Once again, all arguments presented for theformulation (6.42) are valid for (6.51). Both schemes are symmetric in natureas the limiter functions, L(2) and L(3), correct the anti–diffusive term by con-sidering upwind and downwind information (∆uI−1/2, ∆uI+3/2). The schemedefined in (6.42) is referred to as SLIP, or symmetric limited positive scheme,by Jameson [24]. The variation given in equation (6.51) makes the SLIP closeto the class of symmetric TVD schemes [11, 50, 78], which precedes Jameson’sSLIP scheme and adopts a three–parameter family of limiters.

In smooth areas of the solution, the use of either (6.42) or (6.51) willreduce the scheme to a second–order central difference scheme there. Bothsymmetric limiters L(2) and L(3) will be activated if an extremum in the interval[I, I + 1] exists, where the opposite signs of ∆uI−1/2 and ∆uI+3/2 leads toL(2) = L(3) = 0, see figure 6.3. A difference appears in the case of odd–evenmodes where L(3) still returns zero while L(2) will return a limited average of∆uI−1/2 and ∆uI+3/2 as they have the same sign. This is opposite to the signof ∆uI+1/2, as illustrated in the sketch in figure 6.3. Thus, the use of L(2)

will then reinforce the damping of the odd-even modes in a similar way to thefourth–order difference artificial dissipation. This slight difference might havesome influence on the convergence behavior towards a steady–state solution.The non–linear nature of the limiters will introduce some non–linearity as theyare activated and this is one possible explanation for bad convergence behaviornormally faced when using LED schemes without the presence of an additionalfourth–order dissipation in the smooth region of the solution.

Another alternative to the SLIP scheme defined by equation (6.42) isobtained by setting


F (H)I+1/2 =

1

2

(2FI + aI+1/2∆uI+1/2)

−αI+1/2

[∆uI+1/2 − L(2)(∆uI+1/2, ∆uI−1/2)

] (6.52)

if aI+1/2 > 0, i.e. the limited anti–diffusion is computed using solely the upwindinformation. An equivalent expression can be written for aI+1/2 < 0.

(a) (b)

I-1 I I+1 I+2

slopelimitedslope

I-1 I I+1 I+2

slopelimitedslope

(c)

I-1 I I+1 I+2

L (3)

L (2)

Figure 6.3: Effect of limiters for different configurations of an extremum us-ing either L(2) or L(3) computed based on a symmetric support. (a), (b)sign(∆uI−1/2) 6= sign(∆uI+3/2); (c) Odd–even mode.

Supposing aI±1/2 > 0 and using the notation set in (6.44) to (6.46), thesemi–discrete equation for the scheme given by (6.52) can be written as

∆xduI

dt=

1

2

[αI+1/2 − |aI+1/2|

]∆uI+1/2 −

[αI+1/2Φ(1)

(r+I−1/2

)

+ αI−1/2 + aI−1/2 − αI−1/2Φ(1)

(r−I−1/2

)]∆uI−1/2

(6.53)

Taking αI+1/2 = |aI+1/2| in (6.53) one recovers a standard high–order upwindscheme. The term inside the first square bracket is now identically zero and,as aI+1/2 > 0, equation (6.53) reduces to


∆xduI

dt= −∆uI−1/2

2

[Φ(1)

(r+I−1/2

)]|aI+1/2| +

[2 − Φ(1)

(r−I−1/2

)]|aI−1/2|

(6.54)

This upstream limited scheme, USLIP [24], is directly connected to the classof upwind TVD schemes [77, 79]. Consider the USLIP semi–discrete equationfor the point I − 2, but now assuming aI−3/2 < 0 and aI−5/2 < 0 and onceagain α = |a|, which leads to

∆xduI−2

dt= −∆uI−3/2

2

[Φ(1)

(r−I−3/2

)]|aI−5/2|

+

2 −

Φ(1)

(r−I−1/2

)

r−I−1/2

|aI−3/2|

(6.55)

Here the fact that r+I−3/2 = (1/r−I−1/2) and the symmetric condition of the

limiter functions Φ(1) given in (6.45) have been used. To assure LED forschemes (6.54) and (6.55), the limiter Φ(1), and so L(2), has to satisfy the extraconditions

[2 − Φ(1)

(r−I−1/2

)]≥ 0 and

2 − Φ(1)

(r−I−1/2

)

r−I−1/2

≥ 0 (6.56)

The supplementary requirements given in (6.56) can be expressed generi-cally as

Φ(1)(r) ≤ 2 andΦ(1)(r)

r≤ 2 (6.57)

A wide variety of two– and three–parameter limiters, which satisfy therequirements given by properties P1 to P4 mentioned above and the additionalconstraints given by (6.57), exist and will be detailed in section 6.6.

6.4.3 The construction of some high–resolution LED

schemes

Assuming that the requirements presented for the lower–order scheme andfor the limiter functions are satisfied, the procedures described to build flux–limited schemes as a blending of low and high–order schemes, through the con-struction of a corrected flux, is general. Any diffusive scheme, either the first–order upwind schemes presented in chapter 5 or those schemes built with the


explicit addition of artificial viscosity given in section 6.3, can be mixed withany high–order scheme to produce a non–oscillatory high–resolution scheme,in a similar way to that which will be presented for SLIP schemes.

In this section, some aspects of the derivation of the specific schemesstudied here, will be described. The condition to assure LED was alreadydiscussed in last section and no additional proof will be considered for theschemes presented in this section, as they follow the same arguments.

Schemes using Roe flux difference splitting

Lax–Wendroff TVD schemes - The original derivation of Lax–Wendroff TVDschemes presented by Davis [11], and further developed by Roe [50], Sweby [60]and Yee [78], constructs a high–resolution scheme as a sum of a higher–orderscheme (Lax–Wendroff scheme) plus a limited diffusive flux, (6.32), designedin such a way that the final formulation is TVD. With the definition of lim-iters without using the direction of the wave propagation, i.e. symmetric lim-iters, they built for the first time non–upwind TVD schemes. However, in thederivation presented below, the procedure given in equation (6.31) is followed,in order to maintain unified approach to the design of one–step flux–limitedschemes.

The numerical corrective flux F (C)I+1/2, (6.31), for a scalar equation when

Roe’s first–order upwind scheme (5.82) with an entropy fix such as that one

described in equation (5.85) is the low–order flux F (L)I+1/2 and the Lax–Wendroff

scheme [32] is adopted as the higher–order F (H)I+1/2 scheme, would be

F (C)I+1/2 =

1

2

(∆t

∆xa2

I+1/2 − |aI+1/2|)

∆uI+1/2

(6.58)

where ∆uI+1/2 denotes the value of the limited variation obtained after theapplication of the limiter L. The resultant scheme with the use of (6.58)corresponds to the generalised formulation of the TVD Lax–Wendroff scheme(LW/TVD or LW/LED).

Although it might look attractive in terms of CPU requirements, a directextension of (6.58) to system of equations by simply replacing aI+1/2 and uI+1/2

by AI+1/2 and UI+1/2, with the limiter applied on the conservative (or prim-itive) variables, leads to oscillations in the vicinity of strong gradients. Thecomputation of AI+1/2 in Roe’s approach considers a local frozen constantcoefficient system of equations, which allows the use of the transformation de-fined in equation (5.22) and the characteristic variables δW given by (5.24).In this way, the extension of (6.58) to systems of equations can be written inthe form


FFF (C)I+1/2 =

1

2

RI+1/2

(∆t

∆xΛ2

I+1/2 − |ΛI+1/2|)

∆WI+1/2

(6.59)

where the system has been locally decomposed into a set of characteristicequations. The limiter is then applied for each scalar characteristic equationindependently, in which TVD conditions can be verified. The summation ofthe effects of each wave, i.e. coupling, is obtained through the multiplicationby RI+1/2 present in (6.59). This approach is more likely to retain the scalarTVD property and is supported by numerical evidence.

If the limited characteristic variations ∆W are obtained with the use ofa three–parameter limiter L(3), we have a symmetric TVD scheme. On theother hand, if the limiters are computed with two parameters L(2) defined inan upwind–biased way, we have an upwind TVD scheme. Finally, if instead ofusing the Lax–Wendroff scheme as the higher–order scheme, a simple three–point central difference is adopted, the resultant scheme will be referred to asa central difference TVD scheme, which is normally preferred for steady–stateapplications. In reference [78] it is shown that this TVD central differencescheme satisfies the TVD sufficient conditions for most of the limiter functionssuitable for the generalized Lax–Wendroff TVD schemes. The formulationgiven in (6.59), with the variants discussed above, encompasses many TVDschemes developed independently [77].

Second–order limited positive schemes - As already observed, the use of anyof the schemes presented in section 6.4.2 to build a flux–limited scheme willlead to a simple central difference scheme in smooth regions of the solution,despite the fact that the initial high–order scheme (6.41) has an extra fourth–order dissipation term. Jameson [24] then proposes a general approach tobuild higher–order SLIP and USLIP schemes which retain a certain amount offourth–order background dissipation in smooth areas of the solution. In orderto do so, Jameson defines the limited corrective flux F (C)

I+1/2 as

F (C)I+1/2 = sign(AI+1/2) min

(|AI+1/2|, βI+1/2BI+1/2

)(6.60)

where BI+1/2 is a bound determined using a limit of the local gradients. Forinstance, considering the symmetric, SLIP, scheme with a three–parameterlimiter, we have

BI+1/2 = |L(3)(∆uI+3/2, ∆uI+1/2, ∆uI−1/2) | (6.61)

and βI+1/2 is a positive free–parameter which affects the convergence and sta-bility behavior of the scheme. When a general USLIP is devised, it is enough to


change the three–parameter limiter L(3) in (6.61) by the correct local upstreamside two–parameter limiter L(2).

In reference [24], Jameson proves that the final scheme defined by (6.31)using (6.60) is LED, if the lower–order scheme is LED. As discussed in theconstruction of Lax–Wendroff TVD schemes, the limiter procedure must applyon the characteristic variables, which are more like to retain the scalar LEDproperty. The “second–order” SLIP(2) scheme of Jameson when extended forsystem of equations is then constructed by taking

FFF (L)I+1/2 =

1

2

(F I + F I+1) − RI+1/2 |ΛI+1/2|∆WI+1/2

(6.62)

and

FFF (H)I+1/2 =

1

2

(F I + F I+1) − RI+1/2 |ΛI+1/2|

(∆WI+3/2 − 2∆WI+1/2 + ∆WI−1/2) (6.63)

Here the final flux defined by (6.31) uses the corrective flux FFF (C)I+1/2 determined

by

FFF (C)I+1/2 = RI+1/2 |ΛI+1/2| sign( AAAI+1/2) min

(| AAAI+1/2|, βI+1/2 BBBI+1/2

)(6.64)

with BBBI+1/2 computed, using the characteristic variations, as

BBBI+1/2 = |L(3)(∆WI+3/2, ∆WI+1/2, ∆WI−1/2) | (6.65)

The corresponding USLIP(2) is directly defined by considering only the locallyupwind information when computing BBB.

The approach is totally generic and admits any choice for the low– andhigh–order schemes as long as the low–order scheme is LED. The extension tosystems of conservation laws was accomplished using Roe’s FDS but any ofthe upwind generalisations, i.e. FDS, FVS or FD/FV approaches, or even thescalar or CUSP scheme previously described, can be adopted. In particular,if the Lax–Wendroff scheme is taken as the FFF (H)

I+1/2, the SLIP(LW ) representsa variant of the LW/TVD presented in the last section. Jameson [24] alsoshows how to build schemes with order higher than two in the smooth region.However, as it requires a larger stencil of points, the additional cost involvedin its computation and the complexity when extending to multi–dimensionalunstructured algorithms makes such options of no interest in the present study.


Schemes using Osher flux difference splitting

Osher’s first–order scheme [42] for the Euler system of equations, (5.88), com-putes the diffusive term by the summation of integrals computed for eachsimple wave in phase space. Thus, the correct construction of a TVD–likescheme based on Osher’s scheme for systems of equations must apply the lim-iter procedure on each path contribution and later add all the contributions.The simple limiting procedure, without considering intermediate path, leadsto schemes which do not eliminate the oscillations near discontinuities. Forpractical calculations the integrals involved in Osher’s scheme (5.88) are eval-uated in terms of the flux values at the end of each subpath and/or at thesonic points whenever they exist [23]. This represents no extra complication,but requires the application of the limiters on the flux variations rather thanon the characteristic variables.

Γ2

Γ3

I

U1

U2 I+1 I+2I-1

Γ1 Γ1 Γ1

Γ2Γ2

Γ3 Γ3

I-2/3 I-1/3 I+1/3 I+2/3 I+4/3 I+5/3

Γ2<=> u

Γ3<=> u-c

Γ1<=> u+c

Figure 6.4: Integration path and limiter stencil adopted for the constructionof symmetric TVD schemes using Osher’s flux difference splitting.

Assuming a simple central difference scheme as the higher–order schemeand Osher’s first–order upwind as the lower–order scheme, the corrective fluxdefined in equation (6.31) can be written as

FFF (C)I+1/2 =

3∑

i=1

L(3)

[(F I+i/3−1 − F I+i/3−4/3), (F I+i/3 − F I+i/3−1/3),

(F I+i/3+1 − F I+i/3+2/3)]

(6.66)

where the notation is clarified in figure 6.4.

No reference to eventual sonic points is present either in equation (6.66)or in figure 6.4 to avoid extra complexity in the notation. However, it mustbe clear that whenever a sonic point exists, in one of the genuinely non–linear


paths, the flux value at the corresponding intermediate point is replaced bythe flux at the sonic point.

The limiter can be again defined as a function of two–parameters, de-pending on the sign of the eigenvalues at the extremum of each path. If bothextrema have eigenvalues with the same sign, the upwind direction is obviouslydefined, but when they have opposite sign, i.e. when a sonic point is present,there is no unique direction for the wave propagation and a symmetric three–parameter limiter is then used.

6.5 Slope–Limited Methods or Geometric Ap-

proaches

The third general approach for building high–resolution schemes is more ge-ometric in nature and was originated by Van Leer’s [69, 70] observation thatthe first–order spatial approximation of Godunov schemes results from theprojection stage, where the solution is projected in each cell onto piecewiseconstant states. The independence of each step in the Godunov approach, seesection 5.3.2, allows the modification of the projection stage in order to achievehigher–order space accuracy. Thus, the geometric approach can also be calleda generalization of Godunov’s method.

Van Leer [70] adopted a piecewise linear reconstruction using the neigh-bouring cell values, which are subjected to certain monotonic constraints, nor-mally in the form of slope limiters, to avoid under and overshoots in the nu-merical solution. This is the basis for the MUSCL scheme. MUSCL standsfor “monotonic upstream–centered schemes for conservation laws”. Followingthe same lines on Van Leer’s higher–order extension of Godunov methods,Colella and Woodward [8] introduced a quadratic reconstruction which al-lows for higher–order representations than MUSCL. They named their schemePPM or the “piecewise parabolic method”. Another approach, following sim-ilar ideas and which permits even higher–order reconstruction, is the Hartenand Osher [21] ENO methods. The key, and totally unique, idea of adap-tive stencil interpolation permits uniformly higher–order accuracy with sharp,essentially non–oscillatory shock transitions.

When MUSCL–type procedures are applied to the linear scalar equation,in which the physical stage (Riemann solver) can be performed exactly, the ob-tained formulas are easily reinterpreted [33] as flux–limited or algebraic meth-ods, as discussed in previous section. This close connection between the twoapproaches must be kept in mind, as several algebraic schemes can have ge-ometrical interpretation and many geometric schemes can be derived fromalgebraic manipulations.


6.5.1 MUSCL formulation

Using a modified Taylor development [23] it can be shown that cell interfacevalues, up to third–order of accuracy in space, can be evaluated as

uLI+1/2 = uI +

θ

4[(1 − η)(uI − uI−1) + (1 + η)(uI+1 − uI)]

uRI+1/2 = uI+1 −

θ

4[(1 + η)(uI+1 − uI) + (1 − η)(uI+2 − uI+1)]

(6.67)

where superscripts L and R refer to the left and right boundaries of the consid-ered cell. The coefficient θ is included to encompass both first–order schemes(θ = 0) and higher–order schemes (θ 6= 0). Equation (6.67) represents a com-bination of backward and forward extrapolation weighted by the choice of thefree parameter η.

The numerical flux for any of the first–order upwind schemes, summarizedin section 5.5, can be defined according to

F (1)I+1/2 = F(uI , uI+1) (6.68)

where the variables are projected on a piecewise constant basis (θ = 0). Thehigher–order space accurate flux results from using the interface values pro-jected via (6.67), with θ = 1 and defined as

F (H)I+1/2 = F(uL

I+1/2, uRI+1/2) (6.69)

According to the choice of η, different schemes and spatial order of accu-racy is obtained, for instance:

η = −1 ⇒ 2nd–order fully upwind scheme;η = 0 ⇒ Fromm scheme;η = 1/3 ⇒ 3rd–order upwind scheme;η = 1 ⇒ three–point central difference scheme;

In the present work, the linear one–sided extrapolation η = −1 is adoptedand (6.67) reduces to

uLI+1/2 = uI +

1

2(uI − uI−1)

uRI+1/2 = uI+1 −

1

2(uI+2 − uI+1)

(6.70)


The resultant second–order scheme involves a five–point stencil and, al-though it is fully upwind in nature, it possesses similar drawbacks to thoseencountered with centered schemes, i.e. development of oscillations near dis-continuities. Monotonicity principles are invoked at the interpolation stage toprevent overshoots and undershoots in the numerical solution. This is achievedby introducing non–linear limiters so that the interface values are computedaccording to

uLI+1/2 = uI +

1

2Φ+

I−1/2(uI − uI−1)

uRI+1/2 = uI+1 −

1

2Φ−

I+3/2(uI+2 − uI+1)

(6.71)

The reconstruction represented by (6.71)(a) is, in fact, derived from

uLI+1/2 = uI + ∇+

I−1/2(xI+1/2 − xI) (6.72)

where ∇+ is a limited gradient or slope on the cell [xI−1/2, xI+1/2]. If we set

∇+I−1/2 = Φ+

I−1/2

(uI − uI−1

xI − xI−1

)(6.73)

equation (6.72) becomes identical to (6.71)(a) and the flux–limiter Φ can bereinterpreted as a slope–limiter ∇ and vice versa.

As the MUSCL scheme considered here uses only information from theupwind direction during the reconstruction stage, it makes no sense to usesymmetric limiters. Thus, only two–parameter upwind–biased limiters will beused in connection with MUSCL formulations.

The higher–order slope–limited MUSCL scheme is then defined throughthe numerical flux

FI+1/2 = F(uLI+1/2, u

RI+1/2) (6.74)

For instance, let us consider a flux splitting method defined by

FI+1/2 = F+(uLI+1/2) + F−(uR

I+1/2) (6.75)

Assuming the linear advection problem, equation (6.75) reduces to

FI+1/2 = a+uLI+1/2 + a−uR

I+1/2 (6.76)


In order to further simplify, assume a > 0, i.e. a+ = |a| and a− = 0. Thesemi–discretised scheme corresponding to the scheme given by (6.76) can bewritten as

∆xduI

dt= |a|

(uL

I+1/2 − uLI−1/2

)(6.77)

or making use of the definitions of uLI±1/2 given in (6.71), we get

∆xduI

dt= −|a|

uI +Φ(1)

2(r+

I−1/2)∆uI−1/2 − uI−1

+−Φ(1)

2

(r−I−1/2)

r−I−1/2

∆uI−3/2

(6.78)

which can be re–written as

∆xduI

dt= −∆uI−1/2

2|a|[2 + Φ(1)(r

+I−1/2) − Φ(1)(r

−I−1/2)

](6.79)

This equation corresponds to the linear counterpart of equation (6.54). Hence,the limiters here have to satisfy the same conditions discussed in section 6.4.2to obtain an LED scheme. Observe that the symmetry property of the limiterswas used to obtain equation (6.79). For more detailed analysis of the limitersfor MUSCL methods the book of Hirsch [23] is recommended.

The extension of the MUSCL formulation to systems of conservation lawscan be accomplished directly by replacing the single variable u by the vectorU and the limiter Φ to the vector Φ in the above equations. The limiterscan either apply on the conservative variables, primitive variables or mixingboth. Alternatively, using Roe’s approximate Riemann solver, the high–orderMUSCL scheme can still be extended to the system case with the limitersimposed on the characteristic variables. However, this represents an expensivealternative, particularly when multi–dimensional extensions are attempted, asthe characteristic variations must be computed. The use of any of these threesets of variables will be analysed numerically in section 6.9.

Many modifications of Van Leer’s MUSCL scheme have been presentedsince its original presentation, and some of the most important are consideredin Yang and Przekwas comparative study [77]. An alternative approach, inwhich the fluxes in the cell are directly extrapolated to the interfaces andtaken as the numerical fluxes, can be defined in a similar way [23]. However,Anderson et al [2] present some results and arguments which show betterperformance of the variable extrapolation approach when compared to fluxextrapolation methods.


6.6 Non–Linear Limiters: A Compilation

The main mechanism for ensuring a higher-order LED scheme is the use ofnonlinear limiters. The limiters impose constraints on the variations of eitherthe dependent variable or of the flux function. The general conditions imposedon the limiter functions, used in the design of second–order LED schemes, arenow summarized. A detailed discussion of these requirements is given by Sweby[60] and Hirsch [23].

General properties

The conditions presented in equations (6.48) and (6.57), for limiter functionsassociated with a limited–scheme, can be condensed as

0 ≤ Φ(1)(r) ≤ min(2, 2r) (6.80)

which represents the conditions demonstrated by Sweby [60]. The boundedarea given by (6.80) can be represented graphically as given in figure 6.5(a).

(a) (b)

r

1

2φ(r)= 2

(LW)φ(r) = 1

(WB)φ(r) = rφ(r) = 2r

L(1, r)= φ(r)

r1

1

2

21/2

(LW)

φ(r) = 1

φ(r) = 2

(WB)φ(r) = rφ(r) = 2r

L(1, r)= φ(r)

Figure 6.5: TVD regions for the upstream two–parameter upwind–biased lim-ited average functions. (a) TVD region; (b) second–order TVD region.


Assuming, for the sake of simplicity, the linear convection equation witha > 0, the numerical flux of the Lax–Wendroff TVD scheme (6.58) when thelimiter is computed using an upwind stencil, can be written in the form

FI+1/2 =1

2

a(uI + uI+1) − a∆uI+1/2 −

[a2 ∆t

∆x− a

]Φ(1)

(r−I+1/2

)∆uI+1/2

(6.81)

This scheme reduces to the Lax–Wendroff scheme if Φ(1)(r) = 1 and re-duces to the Warming–Beam scheme [74] when Φ(1)(r) = r. These schemesare also plotted in figure 6.5(a), and, as pointed out in [60], both schemes passthrough the point Φ(1)(1) = 1. This is a general requirement for second–orderaccuracy. As argued in [67] any second–order scheme relying on the sup-port [I − 2, I − 1, I, I + 1] represents a weighted average of the Lax–Wendroffscheme and Warming–Beam scheme. This weighted average can be expressed[60] based on the limiter Φ(1)(r) as

Φ(1) = [ 1 − Θ(r) ] Φ(LW )(1) (r) + Θ(r) Φ

(WB)(1) (r) = 1 + Θ(r) [ r − 1 ] (6.82)

The weighting function Θ(r) must represent an internal average,

0 ≤ Θ(r) ≤ 1 (6.83)

as numerical tests [60] using an extrapolation, shows that the resultant schemesare overcompressive. For instance, the adoption of the maximised limiter rep-resented by the upper bound of the region in figure 6.5(a) leads to the exactrepresentation of the solution for the advection of a square wave, but also givesa square wave–type solution for the advection of a sine wave.

The region for the limiter which guarantees second–order upwind TVD,avoiding a resultant overcompressive scheme, is determined using conditions(6.80), (6.82) and (6.83) and can be represented as shown in figure 6.5(b).Then, the construction of a limiter function which lies inside this region willdefine an specific “second–order” limited scheme. It is important to be notedthat as the limiter Φ depends on the data un, the resultant method is non–linear even when a linear problem is considered.

A variety of limiters have been proposed in the literature, which satisfythe properties P1 to P4 given in section 6.4.2 and which lie within the domaindefined in figure 6.5(b). General formulas can be used to derive limiters [24],but here consideration will be restricted to some of the most commonly usedlimiter functions.

The representation of the limiter functions using the alternative notationΦ(1)(r) or Φ(2)(r, s) represents a reduction of one argument when compared


with the equivalent limited average notation L(2)(a, b) and L(3)(a, b, c). In thisway, it simplifies the notation and is helpful for the geometrical representation.However, to avoid in many instances the use of a small parameter to eliminate apossible null value in the denominator or the use of extra logic in the computerimplementation, the limited average L is preferred and will be adopted here.

Two–parameter limiters

Chakravarthy–Osher limiters

L(2)(a, b) = sign(a) max[ 0, min(|a|, βb sign(a)) ] (6.84)

where for a fully explicit upwind scheme 1 ≤ β ≤ 2, [73]. This parameter isdesigned to make the original minimod limiter (β = 1) more compressive. Theexpression (6.84) was proposed by Chakravarthy–Osher [6] and except whenβ = 1 does not satisfy the symmetric property (6.45).

Sweby limiters

L(2)(a, b) = sign(a) max[ 0, min(β|a|, b sign(a)) min(|a|, βb sign(a)) ] (6.85)

where the parameter β is in the range 1 ≤ β ≤ 2, including both extrema ofthe second–order TVD region. If β = 1, equation (6.85) reduces to the simpleminimod function, i.e. the lower boundary of the second–order TVD domain,and if β = 2 we have the upper limit of the second–order TVD domain. Thislast case, was first used by Roe [49] who called it superbee. This gives very goodresults for sharp discontinuities but not so good results for smooth profiles, asit tends to make gradients steeper and to flatten the local maxima.

MUSCL limiter

L(2)(a, b) = sign(a) max[ 0, min(|a|, 2b sign(a))(a + b)

2sign(a)) ] (6.86)

This limiter was first used by Van Leer [70] in his MUSCL scheme, which gavethe name for the limiter. This limiter is also the one adopted by Woodwardand Collela [75] with the formulation written in a slightly different form.

Van Leer limiter

L(2)(a, b) =ab + |ab|a + b + ζ

(6.87)


where ζ is a small constant necessary to avoid possible zero in the denominator.This limiter represents a monotonic increasing smooth function, which was firstproposed by Van Leer [67] in his monotonized Fromm scheme.

Van Albada limiter

L(2)(a, b) =a(b2 + ζ) + b(a2 + ζ)

a2 + b2 + 2ζ(6.88)

This limiter was designed by Van Albada et al [65] in the context of MUSCLschemes and it has similar properties to the Van Leer limiter.

Venkatakrishnan/Thomas limiter

L(2)(a, b) =a(b2 + 2ζ2) + b(2a2 + ζ2)

2a2 − ab + 2b2 + 3ζ2(6.89)

Here ζ is a small parameter, which role is to improve the accuracy near smoothextrema, to reduce the nonlinearity in regions of small gradients and to avoidzero in the denominator [63]. The square of the parameter ζ is determinedas a function of the cube of the grid spacing ∆x. This limiter was used byVenkatakrishnan [71] and by Thomas [63] and its design principles is similarto that of the Van Albada limiter, but it is such that a third–order accurateupwind–biased MUSCL scheme is recovered in the 1–D case in smooth regionsof the flow. In regions of steep gradients, this limiter function falls outside thesecond–order TVD domain (figure 6.5(b)), but still inside the TVD domaingiven in figure 6.5(a).

Three–parameter limiters

The limited average can also be defined with three parameters as arguments.For instance, the Chakravarthy–Osher limiter, (6.84), can be redefined as

L(3)(a, b, c) = sign(a) max[ 0, min(|a|, βb sign(a)), βc sign(a)) ] (6.90)

Performing a similar procedure, three–parameter limiters equivalent to thetwo–parameter limiters defined previously can be designed. An alternativeway to obtain three–parameters limiters [50] consists in defining

L(3)(a, b, c) = L(2)(a, b) + L(2)(a, c) − a (6.91)

where L(2) can be any of the two–parameter limiters. The limiters defined as inequation (6.90) are referred to as non–separable functions, and those given by


(6.91) are called separable functions. The separable limiters may give negativevalues, thus failing to satisfy one of the LED conditions, and non–monotonicresults can be obtained. Also, with the alternation of slope directions, suchas in odd–even modes, function (6.91) will return a negative value and, if nobackground higher–order dissipation is presented in the scheme, we might havestability and convergence problems. These limiters should be avoided and wereincluded here only for completeness of the presentation.

(a) (b)

r1

1

2

2 321

31

Sweby

MUSCL

Cha./Osher

βS

βS

βC

1β

S

βC

L(1, r)= φ(r)

r

1

2

1

VL

VA

V / T

L(1, r)= φ(r)

Figure 6.6: Behavior of some representative two–parameter limited aver-age functions. (a) Non–smooth limiters (Chakravarthy–Osher, Sweby andMUSCL); (b) smooth differentiable limiters (Van Leer, Van Albada andVenkatakrishnan/Thomas).

A sketch of the two–parameter and three–parameter limiter functions ispresented in figure 6.6 and 6.7 respectively. In the figures, the notation definedin equation (6.44) is adopted. Many of the characteristics of limiters which havebeen mentioned previously are clarified with these geometrical representationsof the limiter functions. It can be observed that the limiters in general returnvalues larger than one, which is important to admit larger slopes and oftenleads to better resolution of discontinuities.

It must be observed that the limiters (6.87) to (6.89) are differentiableand that, apart from the Chakravarthy–Osher limiter, all possess the sym-metric property given in equation (6.45). The different characteristics of theselimiters will be demonstrated later through some 1–D and 2–D numerical com-putations.


(a) (b)

-10

12

3 -10

12

3

0

0.5

1

r- r+

φ(r , r )+-

r- r+

φ(r , r )+-

-10

12

3 -10

12

3

-1

-0.5

0

0.5

1

Figure 6.7: Behavior of Chakravarthy–Osher three–parameter limited averagefunction (β = 1). (a) Non–separable limiter; (b) separable limiter.

A limiter will be referred to as an upwind limiter whenever computedusing an upwind–biased stencil as support, and symmetric limiter if this isnot the case. In the local characteristic approach, a scalar scheme on eachcharacteristic field is solved. This adds the flexibility of adopting differentlimiters, and even different supports for the computation of the limiters, foreach different field. Yee [79] suggests that, for problems containing contactdiscontinuities as well as shocks, a mixing of the limiters must be adopted,i.e. the use of different limiters for each characteristic field, which can improvethe overall performance of the scheme. A more compressive limiter should beadopted for the linear fields in order to enhance the contact discontinuities,and for the nonlinear fields more robust limiters might be used in order to getbetter stability and convergence behavior.

6.7 Time–Integration

The solution of both transient and steady–state problems is considered here,via the time dependent form of the equation and by discretising the equationsin space and in time to obtain the numerical solution. In order to obtain a fully


discrete formulation, either a separate spatial discretisation or an integratedtime and space discretisation are considered. The first procedure is oftenreferred to as the method of lines and consists of first discretising in space,reducing the system of PDEs to a system of ODEs, followed by the applicationof any standard numerical method for the solution of ODEs. The secondprocedure considered here is that of Lax–Wendroff type which combines bothtime and space discretisation.

6.7.1 Independent time discretisation

Most of the previously discussed methods, following spatial discretisation, re-duce to a system of ordinary differential equations which may be representedby equation (6.13). This semi–discrete numerical scheme can be expressed inthe general form

dU

dt

∣∣∣∣I

= R(U) (6.92)

where R(U) is the right hand side or residual of equation (6.13), and is totallydetermined for a specific scheme through the definition of the correspondingnumerical flux of the scheme.

Forward difference scheme

The simplest procedure for time integration, already considered in previouschapters, consists in adopting a forward finite difference for the approximationin time. Following this approach, equation (6.92) can be written as

dU

dt

∣∣∣∣I

=Un+1

I − UnI

∆t= R(Un) (6.93)

or

Un+1I = Un

I + ∆tR(Un) (6.94)

where UnI , Un+1

I denote the solutions at node I at time tn and tn+1 = tn +∆t.This method is fully explicit, but with only first–order of accuracy in time andconditional stability.


Predictor–corrector scheme

An alternative procedure can be constructed, using the concept of predictor–corrector schemes [22, 23]. There exist many two–stage methods which canbe considered as belonging to this class of schemes. The scheme adopted hererepresents an iterative approach in which the first, predictor, step determinesintermediate values U after a propagation over half time interval ∆t/2, byusing the explicit time integration,

UI = UnI +

∆t

2R(Un) (6.95)

A second, corrector, step again utilizes an explicit time integration and is givenby

Un+1I = Un

I + ∆tR(U) (6.96)

The procedure defined by (6.95) and (6.96) is considered in the contextof second–order upwind schemes based on the MUSCL variable extrapolationprocedure. In this case, R(Un) of equation (6.95) is determined using thefirst–order numerical flux (6.68). The interface values are now computed as anextrapolation to the intermediate values U, i.e.

ULI+1/2 = UI +

1

2(UI − UI−1)

URI+1/2 = UI+1 −

1

2(UI+2 − UI+1)

(6.97)

which are used to compute the high–order numerical flux (6.69) that deter-mines R(U) in equation (6.96).

The simple explicit time integration (6.94), when applied to the semi–discretised upwind schemes of second–order in space, results in linearly uncon-ditionally unstable schemes, with the instability arising from low–frequencyerrors [23]. The predictor–corrector approach not only stabilises the schemesbut also leads to second–order accuracy in time.

Multistage schemes

A practical solution algorithm can be produced by further discretising the timedimension, utilising an explicit hybrid K-stage timestepping scheme which is ofRunge-Kutta (R-K) type [23, 28]. Multistage schemes represent an important


family of time marching schemes, consisting of K simple explicit stages. Thesolution is advanced over the time–step ∆t according to

U(0)I = Un

I

...

U(k)I = Un

I + αk ∆t R(k−1)I k = 1, . . . , K

...

Un+1I = U

(K)I

(6.98)

where R(k−1)I represents the right hand side of equation (6.13) at stage k and

with the coefficients αk being constants which depend upon the value of Kbeing employed. These coefficients are normally chosen to get an enhancementof the stability range when steady–state computation is considered, and to getthe highest accuracy in time for time–dependent problems [22]. The values ofthe parameters αk adopted here are [23, 46]

3 stages ⇒ α1 = 3/5; α2 = 3/5; α3 = 14 stages ⇒ α1 = 1/4; α2 = 1/3; α3 = 1/2; α4 = 15 stages ⇒ α1 = 1/4; α2 = 1/6; α3 = 3/8; α4 = 1/2; α5 = 1

The possibility for using larger time–steps and the possibility of computingsome contributions to the right hand side R at few stages and freezing themthrough the others results in an efficient computational procedure [25, 27].

6.7.2 Lax–Wendroff combined space–time discretisation

Lax–Wendroff schemes [32] are derived by expanding U(x, t + ∆t) in a Taylorseries, up to second order, as already described in chapter 4, according to

U(x, t + ∆t) = Un+1 = Un + ∆t∂U

∂t

∣∣∣∣n

+∆t2

2

∂2U

∂t2

∣∣∣∣n

(6.99)

With the help of the differential equation, the time derivatives above can beexpressed in terms of space derivatives as

∂U

∂t= −∂F

∂x

∂2U

∂t2= − ∂2F

∂x∂t=

∂

∂x

(A

∂F

∂x

) (6.100)


After substituting into (6.99), we obtain

Un+1 = Un + ∆t∂F

∂x

∣∣∣∣n

+∆t2

2

∂

∂x

(A

∂F

∂x

) ∣∣∣∣n

(6.101)

Following a discretisation with a second–order central difference approx-imation for the space derivatives, the one–step explicit Lax–Wendroff schemeis obtained. Alternatively, one can use the classical Galerkin weighted resid-ual method, see section 2.3.2. The resultant scheme represents the so calledTaylor–Galerkin method, already referred to in chapter 4. The main differencebetween this scheme and the Lax–Wendroff scheme for 1–D problems consistsof the presence of the consistent mass matrix. After lumping the mass ma-trix, an explicit scheme is obtained. Through the use of an iterative proceduresimilar to that given in equation (3.22), the better accuracy obtained with theconsistent mass matrix for transient solution can be recovered, while retainingthe explicit nature of the time marching procedure. The utilisation of a con-sistent mass matrix has the adverse effect of reducing the CFL linear stabilitycriterion [43]. In some of the sketches of 1–D Lax–Wendroff type transientsolutions presented in this thesis, three passes of the iterative procedure areadopted. For further details on accuracy, stability and efficiency of the differentalternatives for the mass matrix, references [12, 43, 82] are suggested.

The Lax–Wendroff combined space–time discretisation procedure, apartfrom stabilizing the explicit centered scheme, possesses second–order accuracyin space and time. A two–step variant, which is more efficient in terms ofcomputational cost was introduced by Richtmyer and Morton [47] and madepopular after the alternative version due to MacCormack [37]. It should be em-phasized that the steady–state solution is dependent on the time–step adopted,and so the Lax–Wendroff approach is normally preferred for transient calcula-tions.

6.7.3 Stability analysis

Linear stability of the full discretised schemes obtained using each of the vari-ous time discretisations techniques, can be performed with the help of methods,such as the Von Neumann method, the equivalent differential equation methodor the matrix method [1, 22, 58]. However, linear stability is not enough, and inparticular when discontinuities are present in the solution a stronger non–linearstability such as TVD–stable sufficient conditions are required. This leads to aCFL–like stability condition which is in general more restrictive than the linearstability limit, and so once satisfied it normally guarantee linear stability ofthe linearised counterpart of the scheme analysed.

Some aspects of total variation stability will be consider here. In refer-ence [33], Le Veque shows that TV–stability of a consistent and conservative


numerical scheme is enough to guarantee convergence. One way to ensure TV–stability is to require that the scheme is TVD and some additional sufficientconditions to complement the conditions presented for the semi–discretisedequation are necessary.

Referring to the semi–discrete scheme given in (6.54) but now considering,for simplicity, the linear counterpart and the fully explicit discretisation (6.94),we get

un+1I = un

I −

∆uI−1/2

2

(a∆t

∆x

) [Φ(1)

(r+I−1/2

)+ 2 − Φ(1)

(r−I−1/2

)]n

(6.102)

In section 6.2.3 the TVD sufficient conditions for the semi–discrete schemewritten as (6.9) is presented. The extra requirement for the fully discretisedscheme, referring to the form given by (6.9), can be stated [18, 23, 60, 78] as

0 ≤ BI I+1 + BI+1 I ≤ 1 (6.103)

The non–negative bound of the sum of the non–linear coefficients BI I+1 andBI+1 I is automatically satisfied from the requirements discussed for the semi–discrete scheme. The other inequality in (6.103) must be further analysed.Since the limiters Φ(1) are restricted to the range given by (6.80), the worstsituation for scheme (6.102) to satisfy (6.103) is obtained when Φ(1)(r

−I−1/2) =

0, i.e.

C2

[Φ(1)

(r+I−1/2

)+ 2

]≤ 1 (6.104)

or assuming that the upper bound of the adopted limiter is represented by ϑ,equation (6.104) can be re–written generically as

|C| ≤ 2

2 + ϑ(6.105)

For instance, the minimod limiter has ϑ = 1 and so |C| ≤ 2/3, while thesuperbee limiter (ϑ = 2) has a more restrictive bound given by |C| ≤ 1/2.

The CFL–like condition given in (6.105) is the same as that obtainedfor the non–linear problem, if the arguments presented for the linear caseare followed. For the TVD Lax–Wendroff scheme, this condition is also valid.This condition can be directly generalised for the Θ–scheme of time integrationleading to

|C| ≤ 2

(2 + ϑ)(1 − Θ)(6.106)


which includes the explicit scheme (Θ = 0) and the unconditionally TVD–stable fully implicit scheme (Θ = 1).

In order to retain the TVD (LED) property for multistage schemes, eitherpredictor–corrector or Runge–Kutta type schemes, a similar CFL–like condi-tion can be derived. Shu [53] derived the CFL requirements that the Runge–Kutta type scheme must satisfy to keep the TVD property. The author’sexperience with the use of Runge–Kutta type schemes with TVD schemes wasnot successful for steady–state applications, in the sense that basically thesame CFL limit as that required for simple explicit scheme was necessary. Shu[53] shows some results, for the class of schemes analysed by him, which givesignificant saving in the computational cost. This option could be investigatedin the future.

6.8 Procedures to build high–resolution schemes

The main steps involved in the methodologies discussed for the design of high–resolution shock–capturing schemes are now presented.

Artificial viscosity schemes

1. Select a second–order centered scheme;2. Build a discontinuity detector or switch;3. If a higher–order diffusion term is not present, add artificially

a high–order diffusion to the smooth region of the solution;4. Add a selective amount of low–order diffusion, originating from any

first–order monotonic scheme;5. Extend the semi–discrete scheme to systems of equations using one of

the possible upwind generalizations;6. Select a time marching method.

Limited schemes

1. Select a first–order monotonic scheme;2. Extend the numerical flux to second–order through the addition

of anti–diffusive flux or through MUSCL reconstruction;3. Correct the terms involved in the previous step using non–linear limiter

functions;4. Extend the semi–discrete scheme to systems of equations using one of

the possible upwind generalizations;5. Select a time marching method;


Remark 1 - It should be kept in mind that, some of the characteristics ofthe first–order scheme are inherited by the final scheme. However, this cannotbe proved and a check, for instance, on the entropy condition is recommended;

Remark 2 - The extension of the scalar scheme for systems of equationsis achieved using either FD, FV or FD/FV splittings or using scalar or CUSPartificial viscosity schemes. The extended scheme cannot rigorously be provedLED;

Remark 3 - A verification of the additional conditions necessary for thefully discretised scalar scheme to retain LED properties must be fulfilled;

Remark 4 - One further step must be considered when multi–dimensionalapplications are attempted. One way to extend the high–resolution schemesto multi–dimension will be detailed in the next chapter;

Remark 5 - Apart from any theoretical verification, a final judgementof the performance of the developed scheme is only possible through extensivenumerical validation.

Remark 6 - In the MUSCL approach, the differencing is followed bythe flux splitting. This makes it easy to implement MUSCL with any split-ting scheme, either FDS, FVS or FD/FV approaches. A simple and commonsubroutine to pre–process the data, projecting and limiting the variables, islinked to an “existing” first–order algorithm. In the algebraic approach, theflux splitting is followed by the differencing and most of the steps necessaryfor the higher–order extension, of the originally first–order algorithm, are de-pendent on each other and specific for each splitting.

6.9 Numerical Study

Some of the shock tube problems utilised in chapter 5, for a comparativestudy of different Riemann solvers, will again be adopted here to address someissues related to the high–resolution schemes detailed throughout this chapter.In addition, a shock tube problem in the supersonic regime, whose solutioncontains the three elementary waves, is also considered during the currentstudy. The initial data for this problem is given by

UL = [1.0, 0.0, 1.0]T for x ≤ 0.0

UR = [0.01, 0.0, 0.01]T for x > 0.0

The shock strength is ≈ 5.392, the maximum Mach number is ≈ 2.406 andthe contact discontinuity moves with velocity ≈ 1.922. The expansion fan is


centered at the original position of the discontinuity. Finally the time for stop-ping the computations was set to Tmax = 0.12, which requires a propagationof approximately 50 time–steps.

Unless it is specifically mentioned to the contrary, a simple explicit forwardtime integration, which is first–order accurate in time, with CFL = 0.5 isadopted.

The assessment of the numerical schemes is centered upon the examina-tion of the accuracy of the predictions. Some issues related to the robustnessand efficiency of the computations are not relevant for the 1–D simulationsperformed here, and will be addressed in the following chapters when dealingwith 2–D calculations.

The present study is by no means an exhaustive investigation. There areadditional elements and parameters in the algorithms themselves, as well asin the physical problems that can affect or interfere with the performance ofthe numerical scheme. However, some of the most important issues, such aschoice of variables to be limited, influence of the time integration, compari-son of limiters and stencil for their computation, entropy fix requirement, areanalysed here. This allows a general insight into the main characteristics ofthe different schemes investigated. In section 6.9.5, a brief comparative studyof the three general approaches to build high–resolution schemes discussed inthis work is also performed.

6.9.1 Choice of variables to be limited

The shock tube problem in the supersonic regime, is adopted to analyse theinfluence of the choice of the set of variables on which the limiters are applied.

In figure 6.8 solutions, in terms of mass flow, for this shock tube problem,when primitive, conservative or characteristic variables are used during thelimiting procedure on the MUSCL/Roe scheme are shown. As expected, thebest behavior is obtained when the high–order schemes, essentially developedand proved LED for scalar equations, are extended to deal with systems ofequations through a locally characteristic structure at the interfaces of thediscrete cells. Despite the attractiveness, in terms of CPU saving, of using theprimitive or conservative variables, they do not always lead to oscillation–freesolutions and are not recommended. It must be observed that the oscillationsreduce to an acceptable level when the shock tube problem in the subsonicregime is considered. This stresses the fact that the performance of a specificscheme is closely related to the flow conditions analysed. In this sense, the useof the characteristic variables plays a major role in terms of the robustness ofthe scheme. Similar conclusions to these, obtained with the MUSCL approach,can be drawn for the algebraic schemes, but this study is omitted here.


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Figure 6.8: Mass flow solutions for the Shock tube problem: supersonic regimecomputed using different set of variables during the limiting procedure on thelimited MUSCL/Roe scheme, using Van Albada limiter. (a) Primitive, (b)conservative and (c) characteristic variables.

6.9.2 Influence of the time integration

In the previous section, the simple explicit Euler time integration was usedin conjunction with the higher–order, MUSCL, spatial discretisation. It is


known [23] that the explicit MUSCL scheme, in the absence of the limitingprocedure, is linearly unconditionally unstable. The limiting procedure sta-bilises the scheme and, depending on the set of variables limited, guaranteesmonotonicity of the solution. Here the influence of the time integration is anal-ysed. Once more, the shock tube problem in the supersonic regime is used andthe set of primitive variables is adopted in the limiting stage of the procedure.

The use of a predictor–corrector method, multistage method or implicitmethod leads to a stable second–order upwind discretisation [23]. When thesetime integration techniques are used with the limited MUSCL/Roe scheme,the results obtained are shown in figure 6.9. As can be seen in this figure,the oscillations present in the solution obtained with the use of the forwarddifference method are either reduced or eliminated. However, this is achievedat an expense of adding too much dissipation to the solution.

It should be remarked that the implicit scheme consists in the use of theSteger and Warming first–order FVS scheme to build the left hand side of thesystem of equations and any of the higher–order schemes can be used on theright hand side. The resultant system of algebraic equations is solved using ablock tridiagonal solver. No detail of the implicit formulation is given here, asit was not extended for 2–D simulation. For an idea of some possible implicitformulations for the class of schemes analysed here, see references [23, 78]. Theutilisation of a spatially first–order implicit operator is only recommendedfor steady–state calculations and a higher–order implicit operator must beused for accurate transient computation. For moderate CFL numbers, theconsistent higher–order implicit discretisation produces high–resolution shockcapturing and nonoscillatory transient solutions [23, 78]. The interest herewas only to demonstrate the influence that the time integration has in termsof damping certain oscillations and not to analyse the performance of thedifferent possibilities of time integration.

6.9.3 Comparison of limiters

Here, the emphasis of the study is focused upon the influence of the adoptedlimiter and the stencil used in its computation. In this study, the CD/TVDscheme using Roe Riemann solver, defined in section 6.4.3, is adopted. Theshock tube problem in the subsonic regime was chosen to perform this study.

In figure 6.10, the use of the upwind–biased limiters is analysed. Whenthe Chakravarthy–Osher limiter is adopted , the compressive parameter β isset to 2.0. All the limiters prove to be accurate in capturing the shock andcontact discontinuity with roughly two to three transition points. However,the behavior of the Roe superbee and MUSCL limiters for the smooth portionof the solution are not adequate, as they are over–compressive, i.e. the gradi-ents of flow variables are made steeper with the formation of a non–physical


expansion-shock, which is clearly apparent in figures 6.10(b) and 6.10(c) re-spectively. A possible remedy for this problem can be the use of the full Lax-Wendroff TVD numerical flux, the adoption of multistage time integration ora reduction in the size of the CFL number [36].

All these options to eliminate the non-physical expansion shock are achievedat the expense of spreading the contact discontinuity and increasing the com-putational time to reach the same time level.

In figure 6.11, the symmetric three–parameter limiters were adopted, whilein figure 6.12 the two–parameter limiters were used. The use of symmetriclimiters in general provides lower resolution of discontinuities, mainly on thecontact discontinuity, as can be seen when the results presented in figure 6.11or 6.12 are compared with those shown in figure 6.10. However, no physicallymeaningless results or stability problems are present.

The separable symmetric limiters, as described in equation (6.91), havemore restrictive CFL conditions. For example, the use of Θ equal to “mini-mod” needs a reduction in the CFL to ≈ 1/2 in order to make it stable, whilethe non–separable limiter works well with the CFL up to 2/3. The results, ingeneral, are similar to those obtained with the use of the corresponding non-separable limiters and therefore the non-separable limiters should be preferred.

In general the use of two or three parameters for the computation of thesymmetric limiter leads to no noticeable difference in the results, as can beobserved when comparing figures 6.11 and 6.12. However, the results usingthe Chakravarthy–Osher limiter, with β set to two, present a small expansionshock on the upper portion of the expansion fan, and nonlinear instability,with oscillations emerging on the constant state between the rarefaction waveand the contact discontinuity.

The small differences in the results are only observed on the smooth portionsof the solutions, as already expected from a theoretical analysis (see figure6.3). In this sense, the three–parameter limiters, which take into account theactual cell variation (∆UIIS

), appear to be more robust and will be used inthe remainder of this work.

The adoption of a compressive factor β > 1.0 proves to be very impor-tant to enhance the precision of the contact discontinuity simulated with theChakravarthy–Osher limiter (see figure 6.13).

The mixing of a more compressive limiter for the linear fields and a dissi-pative limiter for the nonlinear fields are tested now. In figure 6.14 we presentthe results obtained by mixing the limiters in this fashion. Since the reso-lutions of the discontinuities do not always behave in the same way for thedifferent variables, the computed solution for density, velocity, pressure, Machnumber, entropy and temperature are shown here. All flow features are no-ticeably well resolved, with very sharp capture of discontinuities and excellent


representation of the smooth portion of the solution. These impressive resultsmight even be improved with different mixing of limiters and stencils for theircomputations. Nevertheless, it might be argued that this resembles the tuningof parameters present in the artificial viscosity approach, which was criticisedby the author previously. To a certain extent, this argument is true, but thedifference here is that the tuning is supported by a physical interpretation ofthe waves present in the solution and does not represent a purely trial anderror procedure, as with the artificial viscosity schemes.


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Figure 6.9: Shock tube problem: supersonic regime. Computed mass flow, bythe limited MUSCL/Roe scheme, using Van Albada limiter, the set of primitivevariables and different time integration techniques. (a) Forward difference, (b)predictor–corrector, (c) Runge–Kutta (4–stages) and (d) implicit schemes.


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Figure 6.10: Shock tube problem: subsonic regime. Computed density, bythe CD/TVD scheme with limiters computed in an upwind–biased stencil. (a)Chakravarthy–Osher (β = 2.0), (b) superbee, (c) MUSCL and (d) Van Albadalimiters.


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Figure 6.11: Shock tube problem: subsonic regime. Computed density, by theCD/TVD scheme with three–parameter limiters computed in an symmetricstencil. (a) Chakravarthy–Osher (β = 2.0), (b) superbee, (c) MUSCL and (d)Van Albada limiters.


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Figure 6.12: Shock tube problem: subsonic regime. Computed density, by theCD/TVD scheme with two–parameter limiters computed in a symmetricstencil. (a) Chakravarthy–Osher(β = 2.0), (b) superbee, (c) MUSCL and (d)Van Albada limiters.


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Figure 6.13: Computed density for the Shock tube problem: subsonic regime.Study of the effect of the compressive parameter β when Chakravarthy–Osherlimiter, computed in an upwind–biased stencil, is adopted. (a) β = 1.0 and(b) β = 2.0.


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6.9.4 Entropy fix

The shock tube problem defined in chapter 5, whose solution consists of a rar-efaction wave with a sonic point, is chosen to analyse the entropy issue whenhigh–resolution schemes are employed. In figure 6.15, the CD/TVD schemeusing Roe Riemann solver with Chakravarthy–Osher limiter (β = 1.0), i.e.minimod limiter, was used to compute the solution. No entropy correctionwas adopted and the presence of the discontinuity at the sonic point is evidentin figure 6.15(a), in which the limiter is computed in a symmetric stencil. Infigure 6.16(b), the same limiter is now computed using an upwind–biased sten-cil and only a small discontinuity is noted at the sonic point. In figure 6.16(a),the same scheme, but now using the MUSCL limiter calculated with a sym-metric stencil, was used to compute the solution. No unrealistic discontinuityis present in this solution, which is smooth through the sonic point. It can beconcluded that the limiting procedure is not necessarily enough to avoid thepresence of entropy violating discontinuities, but depending on the limiter andon the stencil used for its computation, the sonic discontinuity can be reducedor even eliminated from the solution.

In figure 6.16(b), the solution computed by the CD/TVD scheme, usingthe MUSCL limiter function computed with a symmetric stencil and the OsherRiemann solver, is shown. It is interesting to observe that a small rarefactionshock still persists, even with the limiting procedure which was able to elim-inate this problem when using the Roe Riemann solver. Therefore, even forE–schemes some form of entropy fix might be necessary whenever a sonic pointis present on a rarefaction wave.

The solutions using the CD/TVD/Roe scheme with the minimod limitercomputed in a symmetric stencil, but now with the use of Harten’s entropycorrection (δk = 0.9) and the mixed entropy correction (δ′ = 0.2), defined inchapter 5, can be seen in figure 6.17(a) and 6.17(b) respectively. The use ofany of the two fixes mentioned eliminates the sonic glitch without deterioratingthe accuracy of the solution anywhere else, for this specific problem.

Similar conclusions can be drawn when the limited MUSCL schemes areadopted. When the switched artificial viscosity, scalar or CUSP, approachis employed, the amount of diffusion, added through the tuning of the free–parameters µ(2) and µ(4), necessary to avoid the presence of oscillations in thesolution, is also enough to guarantee an entropy satisfying result. Thus, noneed for an specific treatment of sonic points is necessary. However, when thematrix artificial viscosity approach is used, in which Roe’s matrix is required,a fix such as that discussed previously must be used.

It should be mentioned that at stagnation points, where the linearly de-generate eigenvalue of the Jacobian matrix is null, a similar treatment as that


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Figure 6.15: CD/TVD density solutions for the Shock tube problem: rar-efaction wave with a sonic point using the Roe approximate Riemann solverand minimod limiter. (a) Limiter computed with a symmetric stencil and (b)limiter computed with an upwind–biased stencil.

described for the sonic points is also necessary, and it seems better to refer toa correction for stationary and slowly moving waves.

6.9.5 High–resolution schemes: comparative study

The computed solutions for the subsonic shock tube problem, in terms of thedensity distribution, when the switched artificial approaches are adopted, areshown in figure 6.18. The shock representation is almost the same for all threeschemes analysed. The contact discontinuity and expansion fan are noticeablybetter captured by the matrix–based approach. The second best performance,for this problem, is achieved with the use of the flux–splitting artificial viscosityscheme or CUSP. For all the three schemes, the weighting parameter θ andthe coefficient µ(4) were set to zero, as a low speed transient simulation isperformed. The coefficient µ(2), which controls the 2nd–order diffusive term,


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Figure 6.16: CD/TVD density solutions for the Shock tube problem: rar-efaction wave with a sonic point using the MUSCL limiter computed with asymmetric stencil. (a) Roe Riemann solver and (b) Osher Riemann solver.

was given the values 0.4, 0.8 and 1.3 for the scalar, matrix and flux–splittingapproaches respectively. The presence of a small “staircase” in the solution,at the foot of the expansion fan when the matrix artificial viscosity option isused, can be eliminated either by using some form of background diffusion,µ(4) 6= 0, or any of the second–order time integration procedures discussedhere, as the negative second–order dissipation term embedded in a central–type discretisation will be counter balanced.

Figure 6.19 shows the results computed by the CD/TVD scheme using thesame combination of limiters as that used to compute the results presented infigure 6.14. The use of the Roe or the Osher Riemann solver leads to no notice-able difference on the results. However, the use of schemes which incorporate abackground diffusion, such as the LW/TVD or the SLIP(2) schemes, producesa less sharp resolution of the contact discontinuity and a more rounded leftcorner of the expansion fan in the case of using the LW/TVD scheme, as canbe seen from the plots in figure 6.20. Despite the more dissipative character


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Figure 6.17: CD/TVD density solutions for the Shock tube problem: rar-efaction wave with a sonic point using the Roe approximate Riemann solverand minimod limiter computed in a symmetric stencil. (a) Harten’s entropycorrection and (b) mixed entropy correction.

of these schemes, it will be shown in chapter 7 that the presence of the ex-tra higher–order diffusion might be extremely helpful for a good convergencebehavior towards a steady–state solution in some 2–D applications.

The numerical solution produced using limited MUSCL approaches withdifferent Riemann solvers can be appreciated from the plots in figure 6.21.These plots show roughly the same shock resolution and very similar contactdiscontinuity representation. The main difference appears in the resolutionof the rarefaction wave, in which the solution profile exhibits a small shockon the upper portion of the expansion fan when the AUSM and WPS areadopted. The solution using WPS also shows a kink on the right corner of theexpansion fan. Using always an upwind–biased stencil, as one–side extrapola-tion is adopted in the implemented MUSCL schemes, the mixing of limiterswas attempted. However, no successful combination that surpasses the re-sults obtained using the Van Albada limiter for all the characteristic fields was


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Figure 6.18: Density distributions for the Shock tube problem: subsonic regimecomputed using switched artificial viscosity approaches. (a) Scalar, (b) matrixand (c) flux–splitting (CUSP) schemes.

achieved.


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Figure 6.19: Density distributions for the Shock tube problem: subsonic regimecomputed using algebraic approaches and mixed limiters. (a) CD/TVD withRoe Riemann solver and (b) CD/TVD with Osher Riemann solver.


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0.7

0.8

0.9

1

1.1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(c) (d)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(e) (f)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Comparing the three approaches to build high–resolution schemes, it canbe noted that the MUSCL–based schemes give results comparable to thoseobtained with the algebraic approach, but in general they are a little more dis-sipative. The switched artificial viscosity schemes compare unfavourably to theother upwind–based approaches, mainly in terms of the contact discontinuityrepresentation.


Several high–resolution schemes have been reviewed and applied to solve the1–D Euler system of equations to assess some issues related to shock–capturingschemes and their ability to resolve the flow features present in the solution ofthis set of equations.

It was found that, in order to guarantee the property of robustness of theschemes used to deal with the 1–D Euler system of equations, the extensionof the scalar high-resolution shock–capturing schemes must be performed withthe limiting procedure applied on the characteristic variables. A preliminarystudy of different time discretisation techniques shows the influence that theyhave on the overall solution, but further investigation is required to draw anydefinite conclusion.

The choice of the limiter has been shown to be a crucial component of theoverall performance of the algorithm, exerting a significant influence on theaccuracy of the numerical results, with the limiters computed on an upwind–biased stencil leading to sharper resolution of discontinuities. However, some-times they lead to non–physical features in the smooth portion of the solutionand a more dissipative limiter would be preferred. For steady–state compu-tation, this is not true and the upwind–biased compressive limiters normallyperform better than the symmetric and more diffusive limiters, as will be seenin the next chapters. Mixing the limiters and the stencil for their computa-tion proves very effective when dealing with the algebraic approaches, but thisexperience was not shared for MUSCL–based schemes.

The necessity of judiciously adding a controlled amount of artificial dis-sipation at stationary and slowly moving waves, i.e. at the regions close tostagnation and sonic points, was emphasized. The entropy fix adopted led tothe physically correct solution, while not degrading the accuracy of the solu-tion in the smooth portion of the flow for the problems analysed. The use ofa fix is undesirable as it introduces a parameter that must be retuned for eachapplication and, hence, one of the major advantages of the Riemann–basedschemes over the switched artificial viscosity approach is lost. Despite of thisfact, the use of the fix strategy has proved to be very successful for a number


of applications. The fix is not required in many instances, especially for theE–schemes such as Osher, AUSM, etc.

In terms of accuracy, it was found that the algebraic and geometric ap-proaches give roughly the same resolution, which surpasses that obtained withthe use of a switched artificial viscosity scheme. The use of different fluxsplittings was found not to represent the major element for the accuracyof the scheme for the 1–D problems analysed. However, this is not alwaystrue, as differences still appear in 1–D computations. Furthermore, as themulti–dimension applications will show, mainly when viscous simulation is at-tempted, the Riemann solver option is of paramount importance to obtain aproper result.

The 1–D analysis is helpful to understand many properties of the schemesand to give an insight on the behavior and performance of these schemesfor some elaborate model problems. However, careful investigation of multi–dimensional performance is unavoidable for any decisive conclusion on behalfof any algorithm, either in terms of accuracy, robustness or computationalefficiency.

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Chapter 7

Generalization ofHigh–Resolution Algorithms for2–D Compressible Inviscid FlowSimulation on Triangular Grids

7.1 Introduction

The development of numerical techniques for the simulation of hypersonic vis-cous flows is an area of current practical importance, due to the interest beingshown by the aerospace industry in the development of hypersonic vehicles.The simulation of the hypersonic regimes of interest represents a formidablechallenge to any flow solver due to the highly non–linear nature of the interac-tions which occur. Although our ultimate goal is to simulate numerically thefull system of Navier–Stokes equations, the importance of accurately modellingthe inviscid flow cannot be over emphasized. The success of viscous computa-tions at high Reynolds number is dependent upon how well the scheme succeedsin dealing with the convection dominated character of the flow.

Some of the major difficulties which have to be faced when attempting thenumerical solution of the multidimensional Euler equations are: the non–linearnature and hyperbolic character of the equations; the complex geometries in-volved in industrial applications; the high storage and CPU time involved inpractical simulations.

Over the last decade, CFD practitioners have devoted much effort to thedevelopment and use of unstructured mesh based finite element or finite volumesolution procedures for the compressible Euler equations. These efforts havebeen driven by the promise of rapidly accomplishing the mesh generation for

Compressible Inviscid Flow 258

problems involving complex geometries and the inherent flexibility of the ap-proach, which readily allows for the incorporation of adaptivity. Unfortunately,the unstructured approach is accompanied by some well known drawbacks.The major concerns at present are centered upon the accuracy of the resultscomputed on general grids and the efficiency, in terms of both CPU and storagerequirements. Substantial progress has been made on the development of un-structured mesh methodologies with significant achievements in areas such asmesh generation, adaptivity and the efficiency of unstructured mesh solutionalgorithms, see for instance references [2, 3, 6, 18, 22, 19, 23, 25, 33, 34, 39, 48].However, the dearth of results concerning hypersonic flow simulations usingunstructured meshes has stimulated the present work.

An edge–based data structure finite element approach, first proposed byPeraire et al [38] and which has some features in common with that adopted byBarth [3] and Mavriplis [32], is employed for the representation of the triangulargrid. This allows a direct implementation of different types of standard 1-D upwind or centered shock–capturing methods within an unstructured gridcontext [29, 30, 27, 28, 31, 34, 37]. It is known that the use of this datastructure has additional beneficial effects, in terms of both CPU time andmemory requirements. These thus will be of particular importance when theextension of these methods to the solution of large scale 3–D problems isenvisaged.

In the following sections, a detailed description is given of the numericalformulations employed, with special attention focused on the finite elementedge–based approach and on the extension of the high–resolutions schemespresented in chapter 6 to two–dimensional simulation on unstructured grids.This is followed by the presentation of some remarks concerning improvementsto the basic formulation and implementation considerations. The performanceof the resulting algorithms is assessed by a set of illustrative example solutions.

7.2 A Finite Element Approach

The study of the motion of an inviscid compressible non-conducting adiabaticfluid, in the absence of external source terms, is governed by the Euler equa-tions, see section 2.5.1. This system of equations is considered here in the twodimensional conservation form

∂U

∂t+

∂Fj

∂xj

= 0 in Ω × I for j = 1, 2 (7.1)

where the vector of the conservative variables U and the inviscid flux vectorsF

jare defined in expression (2.59). The equation set is closed by the addition

of the perfect gas equation of state (2.65).


The solution of this set of equations is sought over a closed spatial do-main Ω with boundary surface Γ. The initial/boundary value problem requiresadditionally boundary and initial conditions, which are taken here in the form

Fn

= njFj= F

nat Γ × I (7.2)

and

U(x, t0) = U0(x) on Ω × t0 (7.3)

Here nj denotes the component, in direction xj , of the unit outward normal

vector to Γ and Fn

is the normal flux at the boundary. The exact form of Fn

will depend upon the local solution and the boundary being simulated, while

U0

is assumed to be a known function.

The Galerkin finite element method employing an edge–based data struc-ture described in this work follows the development first proposed by Peraireet al [37, 38, 39].

7.2.1 Approximate variational formulation

An appropriate variational formulation for the problem given by (7.1) to (7.3)can be established as described in section 2.3.1. The first step in the construc-tion of a finite element approximation is the identification of a correspondingdiscrete variational formulation for the problem. Assume that the spatial do-main Ω is discretized into an unstructured assembly of linear triangular ele-ments, with the nodes numbered from 1 to p. The subsets TTT (p) and WWW (p), ofthe trial and weighting function sets TTT and WWW introduced in section 2.3.2,are defined here by

TTT (p) = U(xj , t)|U =p∑

J=1

UJ(t)NJ(xj); UJ(t0) = U0(xj) = U0

J

WWW (p) = W(xj)|W =p∑

J=1

aJNJ (xj)(7.4)

where NJ is the standard linear finite element shape function associated withnode J (located at x = xJ ), UJ is the value of U at node J and a1, . . . , ap areconstants. These sets are conveniently defined for use with a Galerkin finiteelement formulation. An approximate weak variational formulation can thenbe stated as


find U ⊂ TTT (p) such that ∀ t > t0

∫

Ω

∂U

∂tNI dΩ =

∫

ΩF

j(U)

∂NI

∂xj

dΩ −∫

ΓF

nNI dΓ

(7.5)

for each I = 1, 2, . . . , p. The integrals appearing here can be evaluated bysumming individual element contributions, and the compact support of theshape function NI means that the variational statement may be rewritten as

find U ⊂ TTT (p) such that ∀ t > t0

∑

E∈I

∫

ΩE

∂U

∂tNI dΩ =

∑

E∈I

∫

ΩE

Fj(U)

∂NI

∂xj

dΩ −∑

B∈I

∫

ΓB

FnNI dΓ

(7.6)

for each I = 1, . . . , p, where the summations just extend over those elementsE and boundary edges B which contain node I.

To avoid the necessity for numerical integration, a piecewise linear ap-proximation is adopted for the inviscid flux F

jin terms of its nodal values,

i.e. we adopt the representation

Fj(U) =

p∑

J=1

FjJNJ(xj); F

jJ = F

j(UJ) (7.7)

Alternatively, this is equivalent to adopting a Lobatto quadrature rule [17]over each element E when performing numerically the integrals that appear inthe right hand side of equation (7.6). Inserting the assumed forms for U and

Fj(U), described in equations (7.4) and (7.7) respectively, into equation (7.6),

all the integrals present in this equation can be evaluated in closed form. Theleft hand side integral can be expressed as

∑

E∈I

∫

ΩE

∂U

∂tNI dΩ =

∑

E∈I

[∫

ΩE

NINJ dΩ]

dUJ

dt=

[M

dU

dt

]

I

(7.8)

where M is the finite element, consistent, mass matrix. Using the fact thatthe shape function gradients are constant on each element, the integral overthe computational domain, on the right hand side of equation (7.6), can beexpressed as

∑

E∈I

∫

ΩE

Fj(U)

∂NI

∂xj

dΩ =∑

E∈I

[ΩE

3

∂NI

∂xj

]

E

(FjI + F

jJ + F

jK) (7.9)


where ΩE denotes the area of element E with nodes I, J and K. Similarly, theintegral over the computational boundary, on the right hand side of equation(7.6), can be expressed as

∑

B∈I

∫

ΓB

FnNI dΓ =

∑

B∈I

[ΓB

6(2F

n

I + Fn

J )]

(7.10)

where ΓB denotes the length of the boundary edge B with nodes I and J .

The standard finite element data structured consists of the physical coor-dinates simply listed by node numbers, a list of the connectivity of each elementand a list of boundary edges connectivities. With this geometrical and topo-logical data, the integrals discussed above can be performed through a loopover the elements and a loop over the boundary edges with the contributionsto the nodes being accumulated during the process.

7.2.2 Edge–based data structure

As an alternative to the element–based data structure, in which the elementcontributions are sketched in figure 7.1(a), we can represent an unstructuredgrid in terms of an edge–based data structured [3, 32, 38], with the edgecontributions shown in figure 7.1(b). The physical coordinates are simplylisted by node numbers and a list of boundary edge connectivities is adopted,but now the topology inside the domain is characterized through the edgesand their connectivities.

(a) (b)

I

I1

I2

I3

IIm

mEI

2E

1E

I

I1

I2

IIm

I3

S3

S 2

S1

SIm

Figure 7.1: Sketch of the triangles and edges sharing node I, and data struc-tures representation of 2–D unstructured grids. (a) Element–based data struc-ture; (b) edge–based data structure.

For an interior node, such as node I in figure 7.1(b), equation (7.9) canbe rearranged as


∑

E∈I

∫

ΩE

Fj ∂NI

∂xjdΩ =

mI∑

S=1

∑

E∈IIS

[ΩE

3

∂NI

∂xj

]

E

(FjI + F

jIS

)

−∑

E∈I

[ΩE

3

∂NI

∂xj

]

E

(FjI)

(7.11)

where mI is the number of edges in the mesh which are connected to thenode I and the summation

∑E∈IIS

extends over those elements that containthe edge IIS. The second term on the right hand side of (7.11) is zero asit is equivalent to the integration of the gradient of a constant function. Inthis way, expression (7.11) can be evaluated through a sweep over the edgesand by adding the symmetric contribution to the vertices associated with eachedge, as sketched in figure 7.1(b). A similar, but somewhat more elaborate,algebraic manipulation can be done for nodes which lie on the boundary inorder to retain the symmetric nature of the edge contributions, see appendixD. In this way, the discrete equation (7.6) can be conveniently expressed as

[M

dU

dt

]

I

= −mI∑

S=1

CjIIS

2(F

jI + F

jIS

) + 〈2∑

f=1

Df(4Fn

I + 2Fn

Jf+ F

nI − F

nJf

)〉I

(7.12)

where CjIIS

denotes the weight that must be applied to the average value of theflux in the xj direction on the edge S, which joins nodes I and IS, to obtain thecontribution made by the edge to node I. The weight which is applied to thesame quantity to obtain the contribution made by the edge S to node IS willbe denoted by Cj

ISI . In addition, Df represents the boundary face correctioncoefficient which is necessary for nodes I which lie on the boundary and J1,J2 are the two boundary nodes which are connected to node I. These weightscan be readily computed, see appendix D, as

CjIIS

= −∑

EǫIIS

2ΩE

3

[∂NI

∂xj

]

E

+ 〈Γf

6nj

IIS〉IIS

Df = −Γf

12

(7.13)

where the bracketed term is only non zero if IIS is a boundary edge. Thequantity Γf denotes the length of the boundary edge joining nodes I andIS and nj

IISis the component in the xj direction of the unit normal to the

edge IIS. It is readily verified, see appendix D, that these weights satisfy therelations


mI∑

S=1

CjIIS

− 〈2∑

f=1

6DfnjIIS

〉I = 0 for j = 1, 2

CjIIS

+ CjISI = 0 for j = 1, 2 and s = 1, . . . , mI

(7.14)

Once the weight coefficients defined in (7.13) are determined, in a pre–processing stage, the corresponding computer code for the analysis can bewritten so that equation (7.12) is formed by looping over each edge in themesh and sending edge contributions to the appropriate nodes. A loop over theboundary edges is then performed with the extra boundary edge contributionsadded to the appropriate boundary nodes.

For notational convenience, we now define the vector CIISas

CIIS= (C1

IIS, C2

IIS) (7.15)

and let

LIIS= |CIIS

| ; SjIIS

=Cj

IIS

|CIIS| (7.16)

Using these notations we can write equation (7.12) as

[M

dU

dt

]

I

= −mI∑

S=1

LIIS

F IIS︷︸︸︷(F

jISj

IIS+ F

jISSj

IIS)

2

+ 〈2∑

f=1

Df(4Fn

I + 2Fn

Jf+ F

nI − F

nJf

)〉I

(7.17)

From the asymmetry of the edge weights expressed in equation (7.14)(b),the numerical discretization scheme can be immediately observed to possessa conservation property, in the sense that the sum of the contributions madeby any interior edge is zero. It is also apparent, by using the results of equa-tion (7.14), that this is a central difference type scheme. To construct practicalsolution algorithms for the Euler equations we must, therefore, replace the ac-tual flux function F IIS

in equation (7.17) by a consistent numerical flux FFF IIS.

By adopting different forms for this numerical flux function, we are able toconstruct a number of different algorithms for the solution of the compressibleEuler equations [28, 29, 31, 38], some of which will be presented in section 7.3.


7.2.3 Time discretisation

Equation (7.17) represents the time evolution of the unknown vector UI(t)at node I of the mesh. A practical solution algorithm is produced by fur-ther discretizing the time dimension, utilising an explicit hybrid multi–stagetime stepping scheme [21, 39], as already discussed in section 6.7. The multi–dimensional counterpart of expression (6.98), assuming that the nodal valuesUI(t) and RI are known at time tn, is given by

U(0)I = Un

I

...

U(k)I = Un

I + αk ∆t [ML]−1I R

(k−1)I k = 1, . . . , K

...

Un+1I = U

(K)I

(7.18)

where R(k−1)I represents the right hand side of equation (7.17) at stage k. The

values of the parameters αk are given in section 6.7. To reduce the compu-tational cost, a selective multistage scheme [16, 22] can be used, where thedissipative term is only recalculated at certain stages and remains frozen dur-ing the others. To increase the stability range of the explicit multistage timestepping scheme (7.18), and therefore increase the values of the local Courantnumber, a residual smoothing strategy [16, 20, 21, 39] can also be utilised.The simple explicit forward time integration is a particular technique whichcan be derived from the above expression when the number of stages is set toone. The one–step Lax–Wendroff or the predictor–corrector scheme, discussedin section 6.7, extends directly to the multi-dimensional case.

In most applications analysed in this work, a simple explicit, one–stage,time integration is adopted. The reason for such a choice is that little increasein the size of the allowable value of the CFL parameter was achieved whenusing multi–stages in the attempted hypersonic simulations. Further study isrequired in this field and the special class of multi–stage TVD Runge–Kuttatype time discretisations presented by Shu [46] and Shu et al [47] might leadto better performance.

Mass lumping

The consistent finite element mass matrix M is replaced by the standardlumped (diagonal) mass matrix ML. This enables a truly explicit time inte-gration and does not alter the final steady state solution, which is of primary


concern here. Despite some possible loss of temporal accuracy, this approxi-mation was also adopted for the few transient computations performed in thepresent work, see discussion in section 6.7.

Stability

Performing a stability analysis based on the energy method, Giles [10] providesthe following criteria for unstructured mesh cell–vertex algorithms

∆tI = 2C[ML]I

[mI∑

S=1

LIIS|(λmax)IIS

|]−1

(7.19)

This has been written conveniently for the edge–based notation adopted here,where

(λmax)IIS= |uIIS

· SSSIIS| + cIIS

(7.20)

with uIISand cIIS

denoting the edge values of the fluid velocity vector and thespeed of sound respectively. These edge values are obtained by averaging theappropriate nodal values.

Local time–stepping

When a steady-state analysis is studied, a local time stepping [37] is employedto accelerate the convergence rate towards steady-state, since the correct mod-eling of the transient development of the flow is not of interest. This is im-plemented by specifying a constant Courant number C throughout the meshand evaluating the value of the time–step for each node using the relationgiven by (7.19). For true transient simulation, the minimum local time–step,min(∆tI) ∀ I, is adopted for the whole discretisation.

7.2.4 Boundary conditions

Far field boundary faces

For a node I located at the far field boundary, the flux Fn, is determined

by employing Roe’s [41] approximate Riemann solver to resolve the interfacebetween the computed values UI and the free stream values U∞. This meansthat

Fn

I =1

2F n

(UI) + Fn(U∞) − |An(UI , U∞)| (U∞ − UI). (7.21)


where the Roe matrix |An(UI , U∞)| is evaluated here in the direction normalto the boundary using the procedure given in appendix B.

Solid wall boundary faces

At a solid wall, the flux Fn

is set equal to Fn. The nodal normal velocities

are set to zero following each stage of the time–stepping scheme,

u(k) · n = 0 at ΓW (7.22)

This condition is not consistent with the initial data (freestream condition)and an additional complication might appear when attempting to simulatesevere flow regimes which contain obstacles. In fact, truly impulsive start ofany mechanical system is not physically possible owing to “inertia”, and it iseven mathematically inconsistent for incompressible mechanics (fluid or solid),[12], as already mentioned in chapter 4. Rapid start, or impulsive accelerationis quite legitimate and can be implemented by considering that the freestreamcondition is achieved after a certain small time interval. Alternatively, onecan adopt the freestream condition directly, but together with a relaxationon the solid wall boundary condition. This is accomplished here by replacingcondition (7.22) by

u(k) · n = u(k−1) · n(1 − ) at ΓW (7.23)

where is a parameter such that when its value is different from one thesolution will slip and penetrate the wall at the start of the transient, but astime evolves the normal velocity goes to zero at the wall. This procedure hasbeen found to be very important for the simulation of high speed flow pastblunt bodies, where the value = 0.8 has been typically used.

7.3 High–Resolution Schemes for 2–D Unstruc-

tured Discretisations

At present, truly multidimensional upwind–based schemes for the compress-ible Euler equations are still in the research stage of development [5, 42, 50].The available theory is complicated and the implementation for practical ap-plications is currently expensive. The design of multidimensional schemes isnormally accomplished for an arbitrary discretisation, by considering the prop-agation of information locally in the direction normal to the cell faces. Herewith the use of the side–based data structure described in section 7.2.2, weassume that the wave moves in the direction of the weight coefficient vector


CIIS, which is the direction of the centered numerical flux F IIS

. Despite themesh dependence of this approach, the errors involved reduce with the increaseof the order of the scheme, with the high–resolution schemes based upon thisapproach working satisfactorily. This has already been confirmed by many re-searchers and will be demonstrated for the 2–D applications presented in thisthesis.

7.3.1 Conditions to ensure LED property

The hope of obtaining multidimensional high–order TVD schemes through thedirect extension of the 1–D TVD requirements is dashed by the Goodman andLeVeque [11] theorem which states that “Except in certain trivial cases, anymethod that is TVD in two space dimensions is at most first order accurate”.However, for the scalar case the LED property can still be proved [19], as willbe shown next.

Referring to an interior node, a scalar counterpart of the discrete formu-lation (7.12) can be rewritten as

[ML

du

dt

]

I

= −mI∑

S=1

1

2

[Cj

IIS(F j

IS− F j

I )]

(7.24)

in which the property (7.14)(a) was used, i.e. a constant solution 2 × F jI was

subtracted from equation (7.12). Assume a locally one–dimensional propaga-tion of information in the direction SIIS

of the weighting coefficient. As in theapproach followed for the scalar 1–D problem in section 6.4.1, we can definean approximate wave speed aS

IISaccording to

aSIIS

=

(F jIS

− F jI )

uIS− uI

CjIIS

if uI+1 6= uI

∂F j

∂u

∣∣∣∣ICj

IISif uI+1 = uI

(7.25)

Thus, the semi–discrete scheme (7.24) can be written in the equivalent form

[ML

du

dt

]

I

= −mI∑

S=1

1

2

[aS

IIS∆uIIS

](7.26)

which does not satisfy the positivity condition (6.11) required for an LEDscheme whenever aS

IIS> 0. Adding a generic dissipative term defined using

the dissipation coefficient αSIIS

, we get


[ML

du

dt

]

I

=mI∑

S=1

1

2

[(αS

IIS− aS

IIS)∆uIIS

](7.27)

An analogous expression can be obtained if we consider node IS, and the signcondition (6.11) requires that

αSIIS

≥ |aSIIS

| (7.28)

must hold in order that the scheme expressed by (7.27) represents an LEDscheme. However, this scheme is at most first–order accurate and we shouldseek a limited higher–order scheme following the same procedure detailed insection 6.4.2, for 1–D equations. A possible dissipative term of a high–ordernumerical flux can be written as

FIIS= αS

IIS

[∆uIIS

− L(2)(∆u+IIS

, ∆u−IIS

)]

(7.29)

where the differences ∆u±IIS

are defined [19] as

∆u+IIS

= ∇+u · lIIS; ∆u−

IIS= ∇−u · lIIS

(7.30)

Here, lIISis the vector connecting the nodes I to IS and ∇±u are gradients

of u evaluated in the triangles out of which and into of which lIISpoints, as

highlighted in figure 7.2. These reconstructions based on the adjacent trianglesare not unique and further considerations and alternatives will be discussed inthe next section.

It is convenient to introduce the ratio of differences

r+IIS

=∆u+

IIS

∆uIIS

; r−IIS=

∆u−IIS

∆uIIS

; rIIS=

r+IIS

r−IIS

=∆u+

IIS

∆u−IIS

(7.31)

The semi–discrete scheme using (7.29) can then be expressed as

[ML

du

dt

]

I

=mI∑

S=1

1

2

[(αS

IIS− aS

IIS)∆uIIS

− αSIIS

Φ(1)(rIIS)∆u−

IIS

](7.32)

The difference ∆u−IIS

can be generically represented in the form

∆u−IIS

= ǫIαL(uI − uαL

) + ǫIβL(uI − uβL

) (7.33)


α L

βLα R

βR

ιSII

IIL

IS

IR

Figure 7.2: Edge IIS and surrounding triangles.

where the adjacent triangle, represented in figure 7.2 is used to determine thedesired variation. Inserting (7.33) into (7.32) we get

[ML

du

dt

]

I

=mI∑

S=1

1

2

(αS

IIS− aS

IIS)(uIS

− uI)

+ αSIIS

Φ(1)(rIIS) [ǫIαL

(uαL− uI) + ǫIβL

(uβL− uI)]

(7.34)

which is written in the general form given in equation (6.10). Apart from thepositivity condition (7.28) for the lower–order scheme and the non–negativerange of the limiter functions, (6.80), the coefficients ǫIαL

and ǫIβLmust be

non–negative in order to guarantee the LED property of the scheme.

One possible way to determine ∆u−IIS

consists in defining a fictitious node IL asshown in figure 7.2. Using a linear interpolation (extrapolation) of the valuesu at the vertices αL, I and βL, we can compute uIL

. The linear interpolationis represented using the triangle shape functions, but without considering thecompact support normally used in finite element method, i.e.

uIL= NαL

uαL+ NIuI + NβL

uβLwith NαL

+ NI + NβL= 1 (7.35)

The second expression above is required for consistency with a constant fieldu. Combining expressions (7.35) we can write


uIL= NαL

uαL+ [1 − (NαL

+ NβL)]uI + NβL

uβL(7.36)

Therefore, the variation ∆u−IIS

can be approximated by taking

∆u−IIS

= ∆uILI = uI − uIL= NαL

(uI − uαL) + NβL

(uI − uβL) (7.37)

Comparing (7.33) to (7.37), we have that ǫIαL= NαL

and ǫIβL= NβL

. Thelinear interpolation functions NαL

, NβLare, by definition, always bigger than

or equal to zero if IL belongs to the left dashed area, including the boundaries,represented in figure 7.2. The positivity condition given in (6.11) is thenassured for scheme (7.34) and so also the LED property.

Entirely analogue arguments can be used, with the same conclusions asabove, if we consider the node IS instead of I. Furthermore, any of the variousone–dimensional higher–order schemes derived in section 6.4.2 can be adaptedto the unstructured 2–D scalar formulation in the same way we did for thescheme defined by equation (7.29). The extension for systems of conservationlaws can be accomplished following one of the possible upwind generalisationspresented in chapter 5 and 6. As in the case of a non–linear system of equationsin one space dimension, no theoretical guarantee that the resultant scheme willbe LED exists. Despite the ad hoc nature of the multi–dimensional extensionand of the extension for systems of non–linear equations, the numerical schemesdesigned with these assumptions remain second–order accurate for smoothsolutions and, in general, give nonoscillatory and sharp solutions. This canbe observed in the large amount of successful applications performed by manyCFD practitioners and by the results given here.

7.3.2 The Construction of a Local One–dimensional Sten-cil

As already demonstrated in chapter 6 for 1–D problems, any three–point LEDscheme can be at most first–order accurate. Similarly in multi–dimensions, ifonly the surrounding edges to the considered node are used to construct thescheme, it will be at most first–order accurate ( see previous section). Thus,a dependence of the scheme on a bigger support is necessary to build higher–order schemes. For the limited schemes, this is required for the introduction ofthe limiting procedure and for the MUSCL reconstruction. For the switchedartificial viscosity schemes, a bigger stencil is needed for the computation ofthe higher–order background diffusive term and to determine the discontinuitysensor or switch.


A variety of strategies are available for the calculation of the backward andforward differences ∆U±

IISin multi–dimensional unstructured discretisation

[2, 3, 19, 29, 38, 43, 51]. These strategies are in general based on gradientreconstruction and/or interpolation techniques.

The use of a gradient reconstruction

When linear shape functions are adopted in the finite element method, thegradients over each element are constant, with multiple values defined at eachnodal point. One possible way to compute a continuous nodal value for thegradient of the solution is the use of the least squares reconstruction detailedin section 3.2.3. This procedure, when written using an edge–based data struc-ture, leads to an expression similar to equation (7.12), viz

[ML

∂U

∂xj

]

I

=mI∑

s=1

CjIIS

2(UI + UIS

) − 〈2∑

f=1

DfnjIJf

(5UI + UJf)〉I (7.38)

which gives the gradients ∇UI at each nodal point I = 1, . . . , p. Referring tofigure 7.2, the differences ∆U±

IIScan be determined, for instance, if we take

∇U+ = ∇UIS; ∇U− = ∇UI (7.39)

and substituting in equation (7.30). Alternatively one can adopt a centraldifference approximation to the gradients, resulting in the expressions

∆u+IIS

= 2∇U+ · lIIS− ∆uIIS

; ∆u−IIS

= 2∇U− · lIIS− ∆uIIS

(7.40)

The use of these gradient reconstructions for the determination of ∆U±IIS

is very attractive, as it requires little extra memory for the computation. How-ever, the alternatives represented by equations (7.38), (7.39) or (7.38), (7.40)lead to some small oscillations close to shocks or even instability when theyare used in the simulation of some hypersonic flow applications. This lack ofrobustness, found during preliminary computations performed by the author,was later confirmed in the recent study reported by Cabello et al [4]. Differentalternatives are possible [1, 2, 3] which might be considered in a future workwhich searches for a more robust gradient reconstruction procedure.


α L

βLα R

βR

γL

γR

S

IIL

ISIR

Figure 7.3: Location of dummy nodes and determination of triangles for thelinear interpolation.

The use of a linear interpolation

The procedure already discussed when proving the non–negativity of the co-efficients ǫIαL

and ǫIβLin section 7.3.1 constitutes another option to obtain

∆U±IIS

. First, we should introduce the dummy (or ghost) nodes IL and IR asdescribed in figure 7.3. For edge S, the dummy nodes IL and IR are locatedequidistantly along the line which contains the nodes I and IS.

Basically, two choices can be used to obtain the linearly interpolated valuesat the dummy nodes. The first option considers the nodal points that belongto the adjacent triangles to perform the interpolation (extrapolation). Forinstance, the triangle αL, I, βL , shown in figure 7.3, is adopted to computethe value UIL

. The second option considers the actual triangles in which thedummy nodes lie. For instance, now the triangle αL, βL, γL , shown in figure7.3, is adopted to compute the value UIL

. Both options prove to work forthe many problems which have been analysed, with roughly the same final re-sults. However, as argued in appendix C, the use of extrapolation is frequentlytroublesome, and bad behavior was experienced for some hypersonic flow sim-ulations, when the adjacent triangle alternative was considered. If a particularsituation such as that represented in figure 7.4 occurs, either oscillations mightremain in the computed solution or even instabilities can develop, since com-pletely wrong information is used to obtain the variation ∆UISIR

. For thisconfiguration, the use of the adjacent triangle will result in the addition of nodiffusion after the limiting procedure is applied. The use of the actual trianglewhich contains the dummy node IR might add enough dissipation to stabilize


and damp possible oscillations at node IS. Actually, this prove to be truefor all the inviscid and viscous applications analysed during the developmentof this work [27, 28, 30, 29, 31]. As a result, the use of the correct triangleto obtain the information for the interpolation is adopted here. Despite thegood numerical performance, we can no longer prove the LED property, asit is not possible to write the scheme in the general form given in equation(6.10). However, the robustness and the numerical evidences are in favor ofthis option.

adjacent triangle actual

triangle

shockfront

IIS

IL

IR

Figure 7.4: Sketch of a possible shock configuration on a triangular grid.

When a dummy node falls outside the computational domain, differentforms of extrapolation can be used. Whenever one node of the considerededge falls on the boundary, at least one of the dummy nodes will fall outsidethe computational domain, see figure 7.5(a), and either a constant or a linearextrapolation, using the values at the actual nodes of the edge, is adoptedhere to obtain the desired values at the dummy nodes. Despite the localfirst–order nature of the scheme, when a constant extrapolation is adopted,the overall accuracy of the scheme was not deteriorated for the numericalexamples investigated. This choice is normally preferred, as it proves to bemore robust, as the linear extrapolation led to oscillations in some instances.Even when no node of an edge belongs to the boundary, we still can have, inspecial situations, a dummy node outside the domain, e.g. see figure 7.5(b).These special dummy nodes might occur frequently for viscous meshes, but canbe reduced with a proper mesh generation. For these outside nodes, we eitheradopt the constant or linear extrapolation mentioned previously, or considerthe adjacent triangle to the edge for the extrapolation.

Other possibilities to deal with such nodes can be devised and a carefulinvestigation of the accuracy effects should be undertaken, mainly when viscoussimulation is attempted.


(a) (b)

actual nodes

dummy nodes

actual nodes

dummy nodes

adjacenttriangle

Figure 7.5: Dummy nodes outside the computational domain. (a) Edges shar-ing a node with the boundary; (b) Edges without a node on the boundary.

7.3.3 Unstructured grid solution algorithms

The semi–discrete solution algorithm, (7.17), considering a consistent numeri-cal flux function FFF IIS

is written in the form

[M

dU

dt

]

I

= −mI∑

S=1

LIISFFF IIS

+ 〈2∑

f=1

Df(4Fn

I + 2Fn

Jf+ F

nI − F

nJf

)〉I(7.41)

In the present context, we employ the numerical flux FFF IISfor side IIS com-

puted in the direction SSSIIS, (7.16), of the weight coefficient vector. The

superscript S will be adopted to indicate that the quantity is computed in theweight coefficient vector direction.


Once the differences ∆UILIS, ∆UIIS

and ∆UISIRare available, all high–

resolution schemes presented in chapter 6, either using a switch or a limitingapproach, can be directly generalised for multi-dimensions. In what follows,we present a summary of the numerical fluxes FFF IIS

of the schemes extendedfor 2–D unstructured triangular discretisations.

Switched Artificial Viscosity Approaches

The solution algorithm represented by equations (6.14) and (6.18) can be ex-tended for multidimensional unstructured simulation [39] taking the numericalflux function

FFF IIS=

1

2

(F

jISj

IIS+ F

jISSj

IIS) − αS

IIS

[ǫ(2)IIS

∆UIIS

+ǫ(4)IIS

∆UIIS

−(hdU

dl

)

IIS

(7.42)

where the parameters ǫ(2)IIS

, ǫ(4)IIS

have the same definitions presented in section6.3. The values of the edge gradients are taken as the average of the nodalvalues, and given by

dU

dl

∣∣∣∣∣IIS

=1

2

∂U

∂xj

∣∣∣∣∣I

+∂U

∂xj

∣∣∣∣∣IS

dxj

dl(7.43)

where lIISgives the local coordinate at edge IIS, as already shown in figure

7.2.

To compute the pressure switch, equation (6.20), is rewritten here as

ΥI =|pIS

− 2pI + pIL|

(1 − θ)(|pIS− pI | + |pI − pIL

|) + θ(pIS+ 2pI + pIL

) + ε(7.44)

and we use the dummy value of the pressure pIL, determined using one of

the procedures discussed in the last section. For the scaling factor, we canconsider the scalar coefficient αS

IIS= (λmax)IIS

, with (λmax)IIScomputed by

equation (7.20). Alternatively, we can consider a matrix coefficient αSIIS

=

|AS(UI , UIS)|, where the standard Roe matrix between states UI and UIS

isevaluated in the direction of CIIS

, i.e.

ASIIS

= AS(UIIS) =

d

dU

F

jSj

IIS

∣∣∣∣IIS

(7.45)


The flux–splitting artificial viscosity scheme, CUSP, presented for 1–D, isreadily extended for 2–D in a similar fashion as described above, but this wasnot attempted here.

Algebraic Approaches

The schemes discussed in section 7.3.1 are directly generalised and the numer-ical flux functions are now computed in the direction of CIIS

. All variablesand parameters in the following equations have the same meaning as definedfor the 1–D counterpart, see section 7.3.1 for details.

Lax–Wendroff LED schemes [29, 30]

FFF IIS= 1

2 (F

jISj

IIS+ F

jISSj

IIS)

− RSIIS

[λ∗(ΛSIIS

)2∆WS

IIS+ |ΛS

IIS|(∆WS

IIS− ∆W

S

IIS)]

(7.46)

where λ∗ = ∆tS/LS, with LS denoting the length of edge S and ∆tS denotingthe local time step of the edge S.

Second–order limited positive schemes

FFF IIS= 1

2 (F

jISj

IIS+ F

jISSj

IIS)

− RSIIS

|ΛSIIS

|[∆WSIIS

− sign( AAASIIS

)min(| AAAS

IIS|, βIIS

BBBSIIS

)] (7.47)

Galerkin/Osher LED schemes

FFF IIS=

1

2(F j

ISjIIS

+ FjISSj

IIS) − [

∫ UIS

UI

|AS(U)|dU − ∆FS

IIS] (7.48)

where ∆FS

IISdenotes the limited flux variation determined by equation (6.66).

Geometric Approaches (MUSCL)

A straightforward extension of the MUSCL schemes discussed in section 6.5.1can be accomplished for any of the first–oder upwind schemes summarised insection 5.5. To do so, we must consider the limited interface values UL, UR

defined in equation (6.71) and the direction of CIIS.

Roe/MUSCL schemes [27, 31]


FFF IIS=

1

2(F j

(UL)SjIIS

+ Fj(UR)Sj

IIS) − |AS(UL, UR)| (UR − UL) (7.49)

Osher/MUSCL schemes

FFF IIS=

1

2(F j

(UL)SjIIS

+ Fj(UR)Sj

IIS) − [

∫ UR

UL

|AS(U)|dU (7.50)

Liou–Steffen AUSM/MUSCL schemes [28]

FFF IIS=

1

2

MS

L/R[cLΘL + cRΘR] − |MSL/R|[cRΘR − cLΘL]

+ 2pL/R [S1L/R∆1 + S2

L/R∆2] (7.51)

where cLΘL, cRΘR are evaluated with the limited interface values UL andUR, respectively. The interface Mach number MS

L/R and pressure pL/R arecomputed using equations (5.105, 5.108) and (5.104, 5.106), but now using theinterface limited values of the state variables.

7.4 General Remarks

Some remarks concerning practical issues about the implementation of the for-mulations previously described are opportune. The importance of the strate-gies to be discussed is such that they can be responsible for the success orfailure of the analysis. This is particularly true for hypersonic flow simulation.

7.4.1 Enhancement of stability and convergence rate

Certain parameters and elements of the methods analysed can influence thestability and convergence rate of the computation. Some of these parametershave little or negligible effect for low Mach number simulation. However,they can drastically affect the robustness of the scheme when high speed flowsimulation is attempted.


Positivity of Thermodynamic variables

Due to the presence of large gradients, quasi rarefactions zones and impulsivestart from freestream conditions, negative pressure and density can appearduring the time integration path.

To prevent local spurious negative values of the thermodynamic variables ρand p during the convergence process, the pressure and density are updated sothat they always remain positive ([49]), e.g. the pressure is modified accordingto

pn+1 = pn + ∆p[1 + η(α + |∆p

pn|)]−1 (7.52)

whenever ∆p/pn≤α, where η = 2 and α = −0.2.

To increase the robustness of the procedure, for calculations at high Machnumbers and during the initial transient stages of a calculation, following animpulsive start from free stream, the modifications used by Thomas [49] areemployed for the computation of the density and the pressure, which preventsnegative values during the limiting procedure stage. When the Venkatakrish-nan/Thomas limiter, equation (6.89), is adopted with MUSCL schemes, e.g.the limited pressure interface value, which is normally computed as

pLIIS

= pI +1

2L(2)(pIS

− pI , pI − pIL) (7.53)

is now computed using

pLIIS

= pI + L(2)

(pIS

− pI

pIS+ pI

,pI − pIL

pI + pIL

)(7.54)

Similar modification can be used with other limiters, but was not attemptedin this work.

In addition, for some particular cases, normally at the beginning of thetransient, negative values of the square of the sound speed appears when solv-ing the Riemann problem for the dummy edges. One possible explanationfor this spurious behavior could be the extra approximation involved in thecomputation of the dummy node values of the variables and the possibilityfor non–physical states for the downwind dummy node. To circumvent sucha problem, which otherwise will terminate the analysis, we eliminate this par-ticular dummy edge from the limiting procedure. This proves to be particularimportant when a symmetric stencil of points is used as support for the limitingprocedure.


Finally, anyone must be aware of the possibility of negative values of thesquare value of the average sound speed c2

IISwhen Roe’s scheme is adopted

[7, 8, 13], as the average values may lie outside the interval between c2I and c2

IS.

The fix proposed by Einfeldt et al [8] might be necessary, and has been foundto be particularly important for equilibrium real gas simulation by Yee [52].

Background diffusion

The use of the central difference, or Galerkin, type second–order scheme whenbuilding the high–resolution scheme leads to no background diffusion in thesmooth area of the flow solution. On the contrary, the Lax–Wendroff LEDschemes or the SLIP(2) schemes are endowed with a form of background dissi-pation, which proves to be important for helping the schemes damp high fre-quency modes. Therefore, improved convergence rates towards steady–stateare achieved. Of course, this extra diffusion has the drawback of smearing dis-continuities a little and should be well controlled. When the SLIP(2) approachis adopted, we found that the parameter β present in the scheme with a typicalvalue of 0.6 gives good convergence behavior without damaging the accuracy.

Freezing the non–linear artificial diffusion term

The lack of background dissipation and the non–linear nature of the limitingprocedure are believed to be two of the main causes responsible for a badconvergence rate observed for some applications. It seems that the limiter re-acts to small–scale oscillations in smooth regions and thus introduces too muchnon–linearity. Some researchers suggest that freezing the non–linear numericaldiffusion when the solution approaches steady–state helps to drop the residualtowards machine zero. This was confirmed in some numerical experiments inwhich, without the adoption of the freezing strategy, the solution residual wasoscillatory about a constant value after dropping few orders of magnitude. Fora scheme written in the form given in equation (6.31), the freezing strategyis obtained by stopping the update of the limited corrective flux FFFC

IISterm,

which is kept in memory. No considerable difference was found in the solutionwith the adoption or not of a freezing strategy, which was normally imposedafter the L2–norm of the density residual dropped three orders of magnitude.

7.4.2 Improvement of Accuracy

Adaptivity

The unstructured triangulations adopted for the 2-D computations were ob-tained by making use of the advancing front technique due to Peraire et al [40].


An adaptive mesh enrichment procedure for steady state solution was used toimprove the accuracy of the presented computations. The error estimates arebased upon concepts from interpolation theory. In addition, in order to im-prove the efficiency of the refined mesh when used to compute flows which in-volve features with strong directionality, such as shocks, the procedure adoptedalso provides a directional indication of the error. The spatial distribution ofthe “optimal” grid spacing in the direction of each edge is then determinedusing an average of the values of the directional error indicators at the twonodes of the edge. As a result, new nodes, and so elements to keep the gridconsistency, are introduced for each side for which the calculated error exceedsa certain proportion of the maximum error. Further details about the erroranalysis involved in the procedure and about the adaptive procedure itself canbe found in references [26, 34], where several successful results are presentedfor both 2–D and 3–D flow simulations.

It must be noted that the use of an edge–based error indication and refine-ment strategy is also advantageous as it leads to better grid–alignment withthe shocks, in which case the local 1–D basis of the proposed schemes is knownto work undoubtedly well.

7.4.3 The computational implementation

The generated grids, either initial or refined, are provided in the conventionalelement–based data structure format. Thus, a pre–processing of the grid mustbe undertaken before it can be used with an edge–based flow analysis algo-rithm. The pre–processor stage consists basically on the following steps:

1. Build the arrays with the grid and boundary topology, whichare lists of edges and boundary faces with their respective connec-tivities;

2. Compute and store the edges and boundary faces weightingcoefficients;

3. Determine and store the required information necessary for theuse of the dummy nodes;

4. Employ a colouring algorithm [9, 36] to group the edges andboundary faces in such a way that no repetition in the node num-bering occurs amongst items of the same group.

Remark 1 - The information required to describe an unstructured meshis minimal when using an edge–based data structure;

Remark 2 - The three nodes of the triangle that contains the dummynode, and two shape functions evaluated at the dummy node for the interpo-lation step, are kept in memory for each of the two dummy nodes that belong


to each side. This procedure represents a memory overhead of ten times thetotal number of sides in a 2-D computation;

Remark 3 - The colouring algorithm is adopted to prevent recurrenceinside the loops required in the flow algorithms and therefore allowing imple-mentation of the flow algorithms for vector processing.

The operations performed inside the loops over the edges and boundaryfaces, which take place in the flow solver edge–based algorithm are: gatherinformation from the nodes of each edge; operate on this information; scatterit back to the nodes of the edges and add it to the nodal quantities. These typ-ical loops are entirely vectorizable provided each group of edges, or boundaryfaces, is executed separately and a compiler directive instructing the computeris inserted before the vectorizable interior loop. A vectorized version of theG/LED formulation performs more than four times faster than the correspond-ing optimized sequential version on a CONVEX 120 , for a very small problemwith approximately a 1000 discrete points. This performance should improvefor bigger problems, as the vectorized loops are longer.

7.4.4 Further considerations

Some other parameters and techniques which can affect stability, convergencerate, accuracy and efficiency, mainly in high Mach number cases are now men-tioned.

Choice of variables to be limited

In section 6.5.1, we mentioned that our implementation of the MUSCL formu-lations can be extended for systems of equations, with the limiters imposedon the primitive, conservative or characteristic variables. In a preliminarystudy, the use of the primitive variables led to better behavior than the useof the conservative variables or mixing primitive and conservative variables.It was observed in the present study that, depending on the limiter adopted,the choice of the primitive variables leads to good oscillation–free solutions.It is speculated that this behaviour, which disagrees with the 1–D study ofsection 6.9.1, results from the inherent extra dissipation present in the multi–dimensional extension adopted here. This, in a certain sense, contradicts themaxim that 1–D failure will certainly represent a multi–dimensional failure. Asthe MUSCL approach requires some extra computation, during the reconstruc-tion stage, the use of the characteristic variables during the limiting procedurewill make it even more expensive. This fact and the numerical evidences sup-port the choice of primitive variables adopted in this work. However, Yee[52, 53] reports that the choice of characteristic variables plays an important


role in the stability and convergence rate as the Mach number increases, andwe must keep this in mind because the use of primitive variables might leadto failure for some future applications. It was found to be very importantto apply the limiting procedure in the weighting coefficient direction and thenormal direction to this coefficient, when the primitive or conservative vari-ables is adopted. The velocities or momenta are projected onto the weightingcoefficient direction and the normal direction to this coefficient and then thelimiter is applied, followed by another projection back to the x1 and x2 direc-tions. For the algebraic limited schemes, the adoption of any set of variablesother than the characteristic variables leads to solutions which are not freefrom oscillations, even for the 1–D subsonic shock tube application, and thiswas dismissed.

Entropy parameter

The Roe approximate Riemann solver requires the implementation of an en-tropy fix as discussed in section 5.4.1. The value of the free parameter involvedin this fix and/or the different alternatives for scaling this fix is fundamentalin terms of the stability and convergence rate and can even influence the accu-racy of the results. In most of the computations presented here the parameterδ1 of equation (5.85) is set to 0.1, with satisfactory behavior of the scheme.For hypersonic blunt–body calculations Yee [52, 53] and Muller [35] claim thatmore elaborated entropy fix must be used, which might be analysed in futurework.

Multigrid acceleration

The convergence towards a steady state solution of the time marching schemecan be accelerated using a multigrid technique [21, 32, 39]. A multigrid algo-rithm has been recently incorporated by Pang Ling [24], in collaboration withthe author, employing the SLIP2 formulation described previously. The initialresults are encouraging and already demonstrate the importance of the pres-ence of the background diffusion and of the freezing of the non–linear artificialdiffusion inherent in the scheme for good convergence behavior.

Implicit formulation

Since the correct modeling of the transient development of the flow does notrepresent the main concern here, the use of an implicit formulation for the timediscretisation might be very attractive. Due to the scale of the problems nor-mally faced in CFD, iterative approaches might be preferred. An alternative,well suited to be implemented with the unstructured algorithms developed,


consists on the use of the unstructured line relaxation proposed by Hassanet al [14, 15] or the use of nonsymmetric solver such as GMRES [44, 45, 48].Many other alternatives are possible and this represents a huge field of researchto be pursued in a future work.

7.5 Numerical Examples

In this section, the numerical solution of some inviscid flow applications arepresented, to demonstrate the capabilities of the algorithms described earlier inthis chapter. Due to the practical impossibility to test each algorithm for everyimplementation and physical feature, only some of the most important issuesare addressed in this study. A comparative study of the different formulationsdescribed is performed. It must be stated that some of the results which will bepresented here were obtained during the development of this research [30, 29]and so performed before some improvements and alternatives introduced later.In this way, they do not represent the best possible results, but rather a typicalperformance that can be expected from the schemes.

7.5.1 Shock tube problem

The time dependent simulation of the Shock tube problem: subsonic regimedescribed in chapter 5 and extensively analysed through chapters 5 and 6 isadopted as the first application.

Figure 7.6: Mesh Used for a 2-D Simulation of the Shock Tube Problem: sub-sonic regime.

A 2–D simulation of this 1–D problem was performed using the G/LEDand limited MUSCL approaches with Roe approximate Riemann solver. Thecomputational mesh adopted is given in figure 7.6. In order to allow com-parison with the 1–D results presented in section 6.9.5, the same mixture oflimiters is used for the G/LED scheme and the same limiter is adopted for theMUSCL scheme as that used in chapter 6. It must be mentioned that the lim-iting procedure, for the results obtained with the use of the MUSCL scheme,was applied on the primitive variables and not on the characteristic variablesas for the results given for the corresponding 1–D simulation. The reason forsuch a choice is the fact that in most of the following applications the primitivevariables are chosen to apply the limiters for the MUSCL schemes.


(a) (b)

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Figure 7.7: Density distributions for the Shock tube problem: subsonic regimecomputed using the Roe approximate Riemann solver. (a) G/LED using mixedlimiters and (b) Limited MUSCL scheme using Van Albada limiter.

The results, in terms of the density distribution, can be seen in figure7.7where it can be observed that the corners of the expansion fan are slightlymore rounded for the 2–D results. Apart from this, the results are very similar,which demonstrate the good performance of the proposed unstructured high-resolution algorithm for this problem. It was also observed that the adoptionof linear extrapolation for the values at the dummy nodes that fall outsidethe domain (which corresponds to no limiting at that nodes) leads to someoscillations. For this reason, the solution was obtained by making use of aconstant extrapolation, e.g. the first-order upwind scheme is used for thecontribution of the “dummy side” connecting the actual node to the externaldummy node, in this case.


7.5.2 Oblique shock on a flat plate

In the second example, a problem of regular shock reflection at a flat plate isinvestigated. A flow impinging on the plate at a Mach number of 2.0 and atangle of attack of either −100 (flow condition “A”) or +1900 (flow condition“B”) is considered. A regular triangular mesh is used to discretise a rectangulardomain, with 800 elements and 441 nodes and the value of 0.5 is adopted forthe CFL number. The mesh was built in order to provide an almost perfectgrid alignment with the shock for the flow condition with the angle of attackof −100, and the worst, skewed (almost orthogonal), grid alignment when theangle of attack is set to +1900.

(a) (b)

Figure 7.8: First–order upwind density isolines for the Oblique shock on a flatplate using the Roe approximate Riemann solver. (a) Flow condition “A” and(b) flow condition “B”.

The density contours, together with the mesh, for both flow conditionsusing the Roe first–order upwind scheme are shown in figure 7.8. The lostof accuracy of the higher dimensions extension of the standard 1–D upwindmethods when the waves (shock or shear) propagates in a direction far fromaligned with the grid is evident in figure 7.8. In other words, upwinding is notdone in the physical direction of the flow of information, but in the directionwhich is dictated by the mesh. This will introduce a degree of mesh dependencyinto the solution, which can be better appreciated from the results given infigure 7.10(a). However, as already mentioned, this mesh dependence, withconsequent lost of accuracy for some instances, reduces with the increase of


the order of the formulation. This feature can be observed from figure 7.9,where the G/LED scheme was adopted. This scheme is second–order accuratein the smooth portion of the flow, being first–order only in the vicinity of thediscontinuities. The small loss of accuracy when using high–resolution schemescan be further appreciated when comparing the density profiles given in figures7.10(b) with that of figure 7.10(a), obtained with first–order upwind scheme.

(a) (b)

Figure 7.9: G/LED density isolines for the Oblique shock on a flat plate usingthe Roe approximate Riemann solver. (a) Flow condition “A” and (b) flowcondition “B”.

These figures also give an idea of the high resolution achieved by theG/LED over the first-order scheme, where basically only two intermediatenode is seen to be necessary to represent the discontinuity while six is requiredby the first–order upwind scheme, when the “ideal” mesh is adopted.

The possibility of adapting the mesh through the use of a directional errorindicator will produce adapted meshes with good grid alignment, reducing evenmore the worries about mesh dependence. This will be observed in the nexttest cases in which mesh enrichment was employed for the mesh adaptation.

For the Roe first-order upwind scheme no noticeable difference was ob-served in the convergence behavior when either the flow condition “A” or “B”was adopted. Using the G/LED scheme a better convergence during the tran-sient, mainly at the initial stage, was noted when the flow condition “A”, inwhich the mesh is aligned with the shock, was computed. During the laterstage of the transient an opposite behaviour was observed, with the residual


(a) (b)

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

FLOW "A"FLOW "B"

x2

ρ

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

FLOW "A"FLOW "B"

ρ

x2

Figure 7.10: Density profiles at the half domain vertical section for the Obliqueshock on a flat plate using the Roe approximate Riemann solver. (a) First–order upwind scheme and (b) G/LED scheme with MUSCL limiter computedwith an upwind–biased stencil.

going to zero machine only for the case “B” in which the shock is skewed withrelation to the mesh.

Using a discrete computational grid, it is impossible to represent a perfectdiscontinuity unless it lies along the boundary between two mesh cells. There-fore, the discontinuity should ideally be smeared over no more than one to twomesh cells. This simple test case allows an insight into the performance of thehigh–resolution schemes proposed for two–dimensional simulations. Adoptingthe flow condition with angle of attack equal to −100, the results computedusing different schemes, from all three class of high–resolution approaches dis-cussed in this work, are presented in figure 7.11.

The algebraic approaches use the MUSCL limiter computed using anupwind–biased stencil and the geometric approaches use the Venkatakrish-nan/Thomas limiter with the modifications for density and pressure given in


equation (7.54). The free-parameter β present in the USLIP(2) scheme was setto 0.6 for better convergence behavior, but leading to slightly more smearedresult than the other algebraic schemes. The performance of all four differentRiemann solvers together with the limited MUSCL approach is very similar.Any of the algebraic or geometric schemes discussed here achieve a good per-formance with the number of transition points varying from two to four, forthis particular problem configuration. For both switched artificial viscosityschemes the weighting coefficient θ, used in the pressure switch, is set to zeroand the parameter µ(4) is set to 0.1. For the parameter µ(2) the values 0.8 and1.6 were specified for the scalar and matrix scaled schemes, respectively. Noexhaustive study, playing with the values of such parameters, was performedin an attempt to improve the solution. The typical results shown in figure7.11(a) demonstrate the better accuracy obtained with the use of a matrixscaling factor, whose performance is similar to that obtained with the otherhigh–resolution approaches.

Figure 7.12(a) shows the total variation for the characteristic variablesduring the time evolution, and it can be observed that, even with the use of alocal time stepping, the total variation does not grow after the impulsive startfrom the freestream condition. This suggests that the transient solution is alsofree from spurious oscillations.

In figure 7.12(b) the solutions using the different sets of variables for apply-ing the limiters are compared. The MUSCL scheme using the Roe approximateRiemann solver and Van Albada limiter is used. A small pre–shock ripple is ob-served when the primitive or conservative variables are chosen. This might bemore serious for other applications and should be kept in mind. Surprisingly,the worst convergence behaviour is obtained, here, when the characteristicvariables are adopted, with the residual also not going to machine zero.

The convergence history, of the L2-norm for the density residual, Rρ, forthe computations using the first–order Roe upwind scheme, the G/LED schemeand the LW/LED scheme are presented in figure 7.13(a), where a logarithmscaling on the Y axis of the graph is adopted. The observed behavior of theG/LED algorithm is believed to be partially due to the lack of backgrounddissipation when the second order Lax Wendroff term is dropped from thenumerical flux and/or the non-linear nature of the limiting procedure. Thefourth–order numerical viscosity term (background dissipation) damps somehigh frequency modes helping the convergence behavior of the algorithm, butit is not always enough to damp the non–linear effects introduced by the limit-ing procedure and the residual cannot drop beyond a certain level, dependingon the problem, as can be seen from the convergence history of the LW/LEDscheme presented in figure 7.13(a). The convergence behaviour was found todepend strongly upon the choice of the limiter. It seems that certain limitersreact to small-scale oscillations in smooth regions and thus introduce too much


non-linearity. Some researchers suggest that freezing the limiter when the solu-tion approaches steady-state helps to drop the residual towards machine zero.The convergence history, with freezing of the limiters after the L2-norm for theresidual of the conservative variables drops three orders of magnitude for theG/LED scheme can also be observed in figure 7.13(a), where the convergencerate returns to its initial form and the residual drops to machine zero. It mustbe mentioned that at this stage the total variation was already constant, thesolution was already approaching the steady-state, and no considerable differ-ence was found in the solution with the adoption of the freezing strategy, whichcan be seen in figure 7.13(b). Finally, the adopted value of 0.5 for the CFLnumber represents the upper limit according to the non–linear total variationstability criteria given in expression (6.105). If this value is reduced the non–linearity reduces and the residual goes to machine zero, even using the G/LEDscheme. However, this is achieved at an expense of slowing considerably theconvergence rate towards a steady–state solution, which is not desirable.

7.5.3 Flow past a cylinder

The third example consists of a steady flow past a circular cylinder, at a freestream Mach number of 3. A minimod symmetric limiter was adopted in thisanalysis. The presence of sonic, stagnation and rarefaction zones makes thisproblem challenging in terms of stability behavior. The entropy parameter δk

(5.85) was found to exert an important role.

The final mesh, following one adaptation, together with the correspondingMach number contours are shown in figure 7.14. This mesh consists of 24,979elements and 12,651 nodes. Note that both the bow shock and the quasi-rarefaction zone behind the cylinder are well represented, with the circulationand the weak shocks captured.

For the first mesh analysed (not shown here) the adoption of a very smallvalue for δk resulted in bad convergence behavior. For example, the L2-normof the density residual dropped only 2 orders of magnitude after a 10,000 stepsfor δk = 0.02 while it dropped 6 orders, for the same number of steps, if δk isincreased to 0.1. No freezing of the limiters was used. In terms of the solutionobtained, the only difference observed with use of different values of δk wasin the circulation zone behind the cylinder. This reflects the influence of theartificial dissipation added to the linear waves at the stagnation points behindthe cylinder. Further analysis of the influence of the parameter δk can beperformed using different values for δk for different waves “k”.

In this problem, it was found to be extremely important to employ ajudicious choice of the initial mesh, as very coarse triangulation at the backof the cylinder leads to non–physical solutions, which ultimately results infailure of the algorithm, with the computation stopped due to instability. A


reasonable mesh, which enables the capture of the main flow features, is alsoimportant to allow the error indicator to detect regions where the discretemodel must be improved during the adaptation.

The variation in the computed Mach number and pressure coefficient alongthe horizontal symmetry line and over the cylinder are presented in figure 7.15,where the sharp capture of discontinuities is apparent, despite the use of arather diffusive limiter. In figure 7.15(a) we can observe the Mach numberdrop through the bow shock ahead of the circular cylinder, the accelerationover the cylinder wall followed by another drop at the back where it goes tozero, the circulation behind the cylinder and an increase downstream indi-cating the end of the subsonic region. The corresponding pressure coefficientdistribution, with similar features, can be seen in figure 7.15(b). These resultsare in good agreement to that presented in reference [34]. The reduction inpressure to values close to zero at the back of the cylinder leads to negativevalues at the initial transient of the time integration. The procedure describedin equation (7.52) was automatically activated to prevent this from occurring.This would otherwise terminate the analysis. The relaxation of the solid wallboundary condition described in equation (7.23) was found to be importantfor simulations starting from freestream flow condition.

7.5.4 Shock interaction on a cylinder

The previous examples involve computations in a relatively low supersonicregime. The next computation considers a flow at high hypersonic Mach num-ber, in which an oblique shock interacts with the bow shock on a cylinder.The computation starts with the appropriate freestream and oblique shockboundary conditions. Here, the undisturbed free Mach number is 15.0 andthe disturbed flow has a Mach number of 10.596 with 60 angle of attack. Thegeometry definition and flow condition can be seen in figure 7.16.

This is an application with practical interest to the design of hypersonicvehicles [34], as the flow field is typical of what may be experienced by the inletcowl of the vehicles. The values CFL = 0.4, δk = 0.1 and the G/LED using theRoe Riemann solver and the symmetric minimod limiter were adopted. Thechoice of limiter and stencil for this computation is justified by the robustnessof the simple minimod limiter, which gives good stability behavior, and by thefact that, in principle, the intermediate solutions in the adaptive procedureonly drive the procedure and do not need to be very accurate. The initial andfifth meshes analysed with respectively 3,217 and 14,693 elements are shown infigure 7.17. The corresponding Mach number contours are presented in figure7.18.

It should be noted that, even with a very coarse mesh, the Galerkin LEDprocedure resolves the main shock within 2 elements. A stair–case phenomenon


is observed in figure 7.18. The reason behind this phenomenon results fromthe relative direction between mesh and shock. In order to properly resolvethe interaction region in front of the cylinder a finer mesh is needed.

Adaptive mesh refinement appears to be the best choice for the locationof new nodes on the domain in order to enhance the solution accuracy. Al-though the pattern of the flow is significantly more complex than the previousapplications, the convergence rate remains quite satisfactory. The mesh adap-tivity was performed, using the density and velocity field for the computationof the error indicator, after the L2-norm of the density residual drops 5 ordersof magnitude on each mesh, without freezing the limiters. This was achievedafter an average of approximately 10,000 steps, re–starting using the previousmesh solution. The mesh enrichment proves to be extremely important in en-hancing the resolution of the bow shock and also in allowing the capture ofthe shock-on-shock interaction on the front part of the cylinder.

The variation of pressure over the cylinder on the initial mesh com-puted using the G/LED scheme with symmetric minimod limiter and upwindMUSCL limiter are plotted together with the Roe first-order upwind solutionin figure 7.19(a). In figure 7.19(b) the results for the initial mesh using theMUSCL scheme with the Venkatakrishnan/Thomas limiter and different Rie-mann solvers are shown. The solutions using the other Riemann solvers arenot plotted as they are basically the same as the results shown.

The solutions on the final mesh using the same schemes as before arepresented in figure 7.20. It should be observed that the surface pressure at thestagnation point is at least twice as large with the refined mesh, which showsthat all flow features must be captured accurately by the numerical scheme inorder to have a proper surface pressure prediction and once more stresses theimportance of mesh adaptivity.

The pressure solutions obtained using the initial, regular, mesh are smoothwhile that obtained using the adapted mesh show some irregularities. Thisagain reflects mesh dependence. In fact, the quality of the triangles in thefinal mesh (figure 7.17(b)), after five steps of refinement, has deteriorated, ascan be seen in figure 7.21.

This drawback could be avoided if a remeshing procedure was adoptedinstead of mesh enrichment, as more control on mesh quality can be imposed.However, it is impressive that even with such an irregular and low qualitytriangulation the scheme still behaves so well. This allows us to claim that thepresented schemes are robust in dealing with general domain discretisationsfor inviscid flow simulation, at least.

Figure 7.22(a) shows a sketch of the direction through the domain used forthe plot of figure 7.22(b), which shows the Mach number distribution for theinitial and final adapted mesh. The performance of the limited MUSCL scheme


using Venkatakrishnan/Thomas limiter and the Roe approximate Riemannsolver for the capture of the strong bow shock can be evaluated in figure7.22(b). A good shock capture for both initial and final mesh can be noted,and the enhanced resolution when the adapted mesh is used is evident fromthis figure.

Apart from the MUSCL schemes with the Venkatakrishnan/Thomas lim-iter using Roe or Osher Riemann solvers, all the other schemes, using upwind–biased limiters, required a reduction on the CFL number, which in these caseswas taken equal to 0.2. This was necessary not only for better convergencebehavior but also for stability of the computation. Finally, the use of theG/LED scheme with the MUSCL limiter computed using an upwind–biasedstencil lead to an alarming change in the position of the stagnation point. Thisbad behavior shows up when the pressure distribution is examined. I am notsure about the reason for this problem, but one possible explanation can be theovercompressive nature of such a limiter, which was already observed for 1–Dsimulations, and which might in some way damage the solution. The MUSCLapproach was found to be more robust for the simulations performed for thisapplication. However, other parameters, such as the limiter choice, supportof limiters for the geometric approach, CFL number, addition of backgrounddissipation, choice of variables to be limited, etc are some important param-eters which drastically influence stability and convergence behavior. Theseparameters and elements of the formulations must be further analysed for anydefinite conclusion.


The production of high-resolution algorithms for the solution of the compress-ible Euler equations on general unstructured triangular meshes has been de-scribed. Apart from the storage advantages associated with the use of theedge-based data structure, specially for the 3–D extension of the formulation,the computational implementation of a solution algorithm has been found,[39], to require less CPU time for a simulation than the implementation ofthe same algorithm using the conventional finite element data structure. Fur-thermore, the implementation of the described high–resolution schemes, to beused with a generic triangular discretisation, relies on the availability of theedge–based data structure and is directly extendible for 3-D problems on tetra-hedral meshes. The combination of a Galerkin finite element procedure witha rational way of supplying additional numerical dissipation by means of anLED–like limiting procedure proves to be successful in producing accurate androbust algorithms. The flexibility of adapting the mesh and the inclusion ofa mechanism to prevent the appearance of negative thermodynamic variables


allows enhancement of the computed solution and prevents numerical instabil-ities in regions where the solution has very low values of pressure and density.Several other issues related to the accurate and robust simulation of hypersonicflows have been addressed and the numerical results obtained, including somechallenging supersonic and hypersonic applications, demonstrate the potentialof the present scheme.

The use of a symmetric stencil for the computation of the limiters, ingeneral, leads to the addition of more numerical dissipation as the supportfor the calculation of the limiter function is enlarged. This fact can explainthe better convergence behaviour for some applications when using symmetricsupport for the limiter calculation. However, this is not always true, as theauthor has experienced, for some computations, better convergence behaviourusing the corresponding upwind–biased limiter. This might be attributed tothe use of anti–upwind data by the symmetric limiters, which may sometimeslead to a non–physical propagation of backward information resulting froma sudden change in the slope of the solution. This possibility has adverseeffects on the solution, on the convergence behaviour of the scheme or evenon the stability of the computation, as downwind negative thermodynamicvariables, discussed in section 7.4.1, can develop. Furthermore, the use of asymmetric stencil for the computation of the limiter does not prevent failure ofconvergence towards machine zero in certain applications. On the other hand,the pronounced better resolution normally achieved with the use of upwind–biased limiters for the steady–state computations performed here and the factthat no definite better convergence behaviour of any of the two supports wasobserved lead to the conclusion that the upwind–biased limiters should bepreferred for steady–state simulations. This is further supported by the goodresults obtained with the use of the one–sided MUSCL approach, which ofcourse uses upwind–biased support for the limiters. The mixing of limiters,when characteristic variables are adopted and if the physics of the applicationso justify, can represent a good option. Also, when a transient simulation isperformed, the use of certain limiters when computed using upwind–biasedstencil leads to non–physical solutions, as discussed in chapter 6, and theprevious conclusion in favor of upwind limiters is not shared for the transientcalculations.

It was observed that the inclusion of a mechanism to freeze the limiters orthe inclusion of a background diffusion can be necessary for good convergencebehavior, independent of the support used to compute the limiters. Furtherinvestigation, in terms of accuracy, stability and convergence performance, isrequired in order to analyse the effects of the inclusion of a background diffusionand in order to examine the combined effects of the limiter choice and of theentropy correction adopted.

A preliminary observation showed that the convergence rate is also influ-


enced by the type of grid used for the computation. Shock aligned grids andcoarser discretisations normally led to faster convergence than highly skewedgrids with relation to shocks, and finer discretisations.

A CFL number much smaller than the total variation stability limit wasalways found to be necessary when the AUSM or WPS schemes were adopted.Instability of the computations were more frequently with the use of the Os-her Riemann solver, which occurs normally at the starting of the transient.Restarting from any other solution was used when this happened.

The present implementation of the different schemes is performed in thesame program framework. This allows for the comparison of the relative per-formance of different schemes, but also in some way is a constraint on theperformance of a specific scheme. Furthermore, in these implementations, atthe current stage of this research work, little effort was made for specific im-provements on memory and CPU time requirements. With this in mind, theswitched artificial viscosity approach was found to be by far the most efficientapproach, in terms of CPU time to perform a computation. The algebraicapproach requires at least twice as much CPU time as the switched artifi-cial viscosity approach. The MUSCL approach requires at least three timesthe amount of CPU time used with the switched artificial viscosity approach.Within the switched artificial viscosity schemes, the scalar scaling alternative isapproximately 30% cheaper than the matrix option. For the algebraic schemes,the use of the Osher Riemann solver requires roughly twice as much CPU timeas the use of the Roe Riemann solver. In terms of the geometric schemes, thehybrid FD/FV splitting schemes, AUSM and WPS, are the most efficient withthe Roe splitting requiring approximately 50% more CPU time and the Oshersplitting requiring approximately 90% more CPU time. These numbers wereobtained for small scale computations and using sequential computers, givingsolely a rough idea of the performance in terms of CPU time for the schemesanalysed here.

Further improvements in the computational efficiency represents an im-portant step towards the validation of the present unstructured algorithms forthe solution of industrial inviscid flow simulations. Two open possibilities canbe considered with the inclusion of implicit time integration and/or multigridacceleration techniques. Another important and challenging issue refers to theexploitation of the available parallel computer configurations.

Dynamic boundary conditions and moving boundaries must be imple-mented to expand the applicability of the present schemes for some challeng-ing transient calculations. The inclusion of the remeshing adaptive procedure,similar to that described in chapter 4, should also be considered for an effectivesimulation of this family of problems.

It is well known how important is the treatment of the numerical boundaryconditions, with impact on the performance in terms of accuracy, stability and


convergence behaviour of the algorithm. Further study centered upon thissubject must be undertaken with the hope to improve some characteristics ofthe present formulations.

As already mentioned, some of the studies and results presented here wereperformed before the incorporation of some improvements on the solution al-gorithms and before the implementation of some of the described formulations.A systematic and careful comparative study of all discussed schemes for thepresent applications and other more challenging applications represents an im-portant task to be pursued in the near future.


(a)

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

AV-SCALARAV-MATRIX

x2

ρ

(b) (c)

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

G-LED-ROE G-LED-OSHERLW-LED-ROE

USLIP ρ

x2

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

MUSCL-ROE MUSCL-OSHERMUSCL-AUSM MUSCL-WPS

2x

ρ

Figure 7.11: Density profiles at the half domain vertical section for the Obliqueshock on a flat plate using different high–resolution schemes. (a) Switchedartificial viscosity approaches, (b) algebraic approaches and (c) geometric ap-proaches.


(a) (b)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

Field ( u )

Field ( u )

Field ( u-c )

Field ( u+c )TV

t

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Prim.-var.cons.-var.char.-var.ρ

x2

Figure 7.12: Oblique shock on a flat plate problem. (a) Total variation of thecharacteristic variables for the G/LED scheme, (b) solutions at the half domainvertical section using the MUSCL–Roe scheme with Van Albada limiter anddifferent sets of variables.


(a) (b)

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 100 200 300 400 500 600 700 800 900 1000

1st-UPW ==>

LW-LED

G-LED

<== G-LED (freezing)

Steps

Rρ

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Non-freezingFreezing

ρ

x2

Figure 7.13: Convergence study for different schemes used to solve the Obliqueshock on a flat plate with the MUSCL limiter computed using upwind–biasedstencil, whenever is the case. (a) Convergence history of the L2–norm of thedensity residuals and (b) Solutions at the half domain vertical section usingG/LED scheme with and without freezing the limiters.


(a) (b)


(a) (b)

0

0.5

1

1.5

2

2.5

3

3.5

4

-3 -2 -1 0 1 2 3

M

x1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-3 -2 -1 0 1 2 3

C p

x1

Figure 7.15: Steady flow past a cylinder at Mach number 3. (a) ComputedMach number and (b) computed pressure coefficient on the centre line and onthe cylinder surface.


2.0

3.70

62.

794

α o= 6

α o= 0

2.0

2.0

2.5

M 8 = 10.596

M 8 = 15.00

Figure 7.16: Problem definition for the Shock interaction on a circular cylinderat a Mach number of 15.


(a) (b)

Figure 7.17: Shock interaction on a circular cylinder at a Mach number of 15.(a) Initial mesh and (b) final adapted mesh.


(a) (b)

Figure 7.18: Computed distribution of the Mach number contours for the Shockinteraction on a circular cylinder at a Mach number of 15. (a) Using the initialmesh and (b) Using the final adapted mesh.


(a) (b)

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

1st-order G-LED-Sym.

G-LED-Upw. P

x2

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

MUSCL-ROE MUSCL-AUSMP

x2

Figure 7.19: Computed distribution of the pressure on the surface of the cylin-der for the Shock interaction on a circular cylinder at a Mach number of 15 us-ing different high–resolution schemes and the initial mesh. (a) Roe first-orderupwind scheme, Galerkin LED scheme with the symmetric minimod limiterand the upwind MUSCL limiter and (b) limited MUSCL using Venkatakrish-nan/Thomas limiter and the Roe or the AUSM flux splitting.


(a) (b)

0

2

4

6

8

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

1st-order G-LED-Sym.G-LED-Upw.P

2x

0

2

4

6

8

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

MUSCL-ROE MUSCL-AUSMP

x2

Figure 7.20: Computed distribution of the pressure on the surface of the cylin-der for the Shock interaction on a circular cylinder at a Mach number of 15using different high–resolution schemes and the final adapted mesh. (a) Roefirst-order upwind scheme, Galerkin LED scheme with the symmetric minimodlimiter and the upwind MUSCL limiter and (b) limited MUSCL scheme usingVenkatakrishnan/Thomas limiter and the Roe or the AUSM flux splitting.


Figure 7.21: Detail of the region surrounding the stagnation point in the finaladapted mesh used for the Shock interaction on a circular cylinder at a Machnumber of 15.

(a) (b)

1350

Z

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

InitialFinalM

Z

Figure 7.22: Shock interaction on a circular cylinder at a Mach number of15 using limited MUSCL scheme with the Venkatakrishnan/Thomas limiterand the Roe flux splitting. (a) Direction used for plot and (b) Mach numberdistribution for initial and final meshes.

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Chapter 8

Extension for 2–D CompressibleViscous Flow Simulations

8.1 Introduction

In recent years, researchers have demonstrated that unstructured grid finiteelement and finite volume methods can be successfully employed for the simu-lation of aerodynamic flows. Impressive transonic flow computations involvingcomplex geometries have been made, but there has been little correspondingwork on the simulation of viscous flows and almost no work directed at thearea of hypersonic viscous modelling. Major problems which have to be facedin this area are the construction of suitable algorithms for the generation ofunstructured meshes and the development of robust flow solvers for implemen-tation on such meshes. In this chapter, we are addressing only the problemof flow solver development and we present procedures for the implementationof flow algorithms which can be employed successfully for the simulation ofhypersonic laminar viscous flows on unstructured triangular grids.

Since compressibility is associated with high velocities and therefore highReynolds numbers, the physical problems analysed here are assumed to beinviscid dominated, in the sense that apart from very thin regions close tosolid walls (boundary layers) the flow field is governed by the inviscid portionof the Navier–Stokes equations. As a result, high–resolution shock–capturingschemes are needed to resolve the shock waves and contact discontinuitiespresent in the solution. Since the viscous terms in the Navier–Stokes equationsare parabolic or elliptic in nature, the numerical discretisation of these termsis accomplished by a centered type technique. In conclusion, the numericalprocedures adopted here for the compressible Navier–Stokes simulations arebuilt using any of the schemes developed in chapter 7 to discretise the inviscidpart of the Navier–Stokes equations and a mixed finite element formulationfor the viscous terms of the Navier–Stokes equations.

Compressible Viscous Flow 313

The inherent numerical dissipation, due to the use of a shock–capturingscheme, has in principle a conflicting effect on the physical viscosity in theboundary layer region, where steep gradients of the flow variables exist. How-ever, numerical evidence supports the procedure adopted as the contaminationof the boundary layer by the numerical diffusion mentioned above remainswithin an acceptable limit, and the solution away from the boundary layer isalso accurately simulated.

In the current chapter, three possible ways to extend the Euler solversof chapter 7 to Navier–Stokes are discussed. The alternative which requiresno memory overhead and which is suitable for the use of an edge–based datastructure is adopted and the performance of the schemes is demonstrated bytheir application to the simulation of some steady–state test cases.

8.2 Viscous Fluxes Computation

The equations which govern the unsteady laminar flow of a compressible vis-cous fluid can be written, in the absence of external source terms, in theconservation form

∂U

∂t+

∂Fj

∂xj=

∂Gj

∂xjin Ω × I for j = 1, 2 (8.1)

Here, U is the vector of the conservative variables, while Fj

and Gj denotethe inviscid and viscous flux vectors in the direction xj respectively, which aredefined in expression (2.59). The set of Navier–Stokes equations is closed bythe addition of the stress/rate of strain relationship (2.60), the equations ofstate (2.61) and Sutherland relation (2.62).

The boundary and initial conditions for the inviscid fluxes are given inequations (7.2) and (7.3), and they must be complemented by the viscous fluxboundary condition taken here in the form

Gn = njGj = Gn

at Γ × I (8.2)

where Gn

represents the value of the viscous normal flux to the boundary, andthe exact form will depend upon the local solution and the boundary conditionbeing simulated.

As mentioned previously, independent of the choice of the basic Eulersolution algorithm, the viscous flux terms are always discretised in a centeredway. Following exactly the same procedure employed for the inviscid equation,detailed in section 7.2.1, the left hand side of the weak approximate variationalstatement (7.6) must be complemented with the viscous terms


−∑

E∈I

∫

ΩE

Gj

(U,

∂U

∂x1,

∂U

∂x2

)∂NI

∂xjdΩ +

∑

B∈I

∫

ΓB

GnNI dΓ (8.3)

for each I = 1, . . . , p, where the domain and the boundary integrals wererepresented considering the sum of individual element and boundary facescontributions. The integrals involved in these terms can be evaluated usingdifferent strategies.

8.2.1 Standard finite element approach

The conventional finite element approach considers, for computational econ-omy, a piecewise constant approximation for the viscous flux Gj in terms of asingle element value [11], i.e. we use the representation

Gj

(U,

∂U

∂x1,∂U

∂x2

)=

m∑

E=1

GjEPE ; G

jE = Gj

(U |E,

∂U

∂x1

∣∣∣∣∣E

,∂U

∂x2

∣∣∣∣∣E

)(8.4)

where PE denotes a piecewise constant shape function defined in a similar wayto the weighting functions in the subdomain method given in (2.15). This isequivalent to adopting a one point numerical integration rule over each elementE when computing the integrals of equation (8.3). Observe that ∂U/∂xj is

constant since U, (7.4), is assumed linear, and U |E is a mean element value ofU. After inserting the assumed form for U given in (7.4) and for Gj given in(8.4), the integral over element E with nodes I, J and K, on the right handside of equation (8.3), can be expressed as

∫

ΩE

GjEPE

∂NI

∂xjdΩ = ΩE

∂NI

∂xj

∣∣∣∣∣E

Gj

((UI + UJ + UK

3),

∂U

∂x1

∣∣∣∣∣E

,∂U

∂x2

∣∣∣∣∣E

)(8.5)

In a similar way, the integral over the boundary B, on the right hand side ofequation (8.3), can be expressed as

∫

ΓB

GnEPE dΓ =

ΓB

2G

n

(

UI + UJ

2),

∂U

∂x1

∣∣∣∣∣EB

,∂U

∂x2

∣∣∣∣∣EB

(8.6)

where EB denotes the element adjacent to the boundary ΓB. It should benoted that to compute the assembled form of equations (8.5) and (8.6) anelement–based data structure must be used.


8.2.2 Edge–based finite element approach

Noting that a generic term of the vector Gj , see equation (2.59), has a typicalform f ∂g/∂xj , one alternative way to evaluate the integral of the viscous fluxescan be devised [6]. It is possible to show that the integral of this typical term,for any interior node I, can be written as

∑

E∈I

∫

ΩE

f∂g

∂xk

∂NI

∂xjdΩ =

mI∑

S=1

(fI + fIS)

2

EjIIS

2(gj

I + gjIS

) (8.7)

where the coefficients EjIIS

are determined by

EjIIS

=∑

E∈IIS

ΩE

3

∂NI

∂xj

∣∣∣∣∣E

∂NI

∂xk

∣∣∣∣∣E

(8.8)

Expression (8.7) must be suitably modified for the computation of nodes whichlie on the computational boundary. Once the coefficients Ej

IISand the correc-

tive boundary coefficients are computed and stored in a pre–processing stage,the final expression can be assembled using an edge–based data structure.

8.2.3 Finite element mixed formulation

In a third alternative [13] the nodal values of the viscous fluxes terms aredirectly evaluated as

GjI = Gj

(UI ,

∂U

∂x1

∣∣∣∣∣I

,∂U

∂x2

∣∣∣∣∣I

)(8.9)

with the gradients of the variables at node I computed using the variationalrecovery procedure given by expression (7.38). The viscous flux Gj can be

represented using a piecewise linear approximation, as the inviscid flux Fj

(7.7). In this way, the edge–based data structure can be employed and theNavier–Stokes equations may now be discretised in the same manner as theEuler equations, leading to the expression

[M

dU

dt

]

I

= −mI∑

s=1

CjIIS

2

(F

jI + F

jIS

) − (GjI + G

jIS

)

+〈2∑

f=1

Df

(4F

n

I + 2Fn

Jf+ F

nI − F

nJf

)

−(4Gn

I + 2Gn

Jf+ Gn

I − GnJf

)〉I

(8.10)


which requires no memory overhead and is, of course, suitable for use in anedge–based code.

It must be noted here that the discretization of the viscous terms em-ploying the mixed formulation involves information from two layers of pointssurrounding the point under consideration while the other two approacheswould use information from those points directly connected to the point beingconsidered. However, no adverse effect was observed with the use of the mixedformulation for some subsonic numerical experiments performed by Peraire etal [13]. Similar conclusions were derived from some recent comparisons for hy-personic regimes performed by Manzari [10]. Luo et al [6] found also practicallyidentical results comparing the mixed formulation with the edge–based finiteelement approach given in (8.7). The impossibility of evaluating the standardfinite element expression using an edge–based structured, the significant stor-age overhead required by the alternative represented in equation (8.7) and thesimilar results obtained with all three approaches, motivates the adoption ofthe mixed formulation.

8.3 General Remarks

8.3.1 Boundary conditions

Far Field Boundary Faces

For a node I located at the far field boundary, the flux Gn, is determined

by employing the free stream values at inflow (Gn

= Gn∞) and the computed

values at outflow (Gn

= Gn).

Solid Wall Boundary Faces

At a solid wall, the velocity components are set to zero. However, for thesimulation of high speed flow pasting blunting bodies we adopt the relaxationapproach described in equation (7.23) but now for both the normal and thetangential velocity components. In addition, isothermal or adiabatic conditionsare set. At a point on an isothermal wall, the temperature (or energy) is set tothe prescribed value. At an adiabatic wall, the condition of zero temperaturegradient is weakly imposed through the discretisation of the energy equation.

8.3.2 Stability of explicit time integration

In the absence of a well grounded stability criteria for the full set of Navier–Stokes equations, a criterion derived by analogy with the advection–diffusion


equation is adopted [16]. The resultant expression is closely related to the cri-terion given in reference [17], which was developed based on the spectral radiusof the viscous Jacobian matrices. The local time step for viscous computationis then determined by replacing expression (7.19) by

∆tI = 2C[ML]I

mI∑

S=1

LIIS

|(λmax)IIS| +

4L2IIS

µI max(2;

γ

Pr

)

Re∞ρI [ML]I

−1

(8.11)

This heuristic criterion reduces to the Euler criterion in the limit of vanishingviscosity [µI → 0] and reduces to the pure diffusive limit, either for the momen-tum equation or the energy equation, when the convective term is negligible[(λmax)IIS

→ 0]. The same remarks and strategies employed to accelerate theconvergence rate towards steady–state presented in section 7.2.3 remain validhere.

8.3.3 Wall coefficients

The distribution of the pressure coefficient, heat transfer and skin friction arenormally required either during the stage of validation of the Navier–Stokessolution algorithm or during the use of this algorithm for aerodynamic design.

The wall pressure pw is directly obtained using the flow variables and thestate equation (2.61). The heat flux at the wall qw and the wall shear stressτw are obtained [1], respectively, from

qw =∂T

∂n= ∇T · n (8.12)

and

τw = t2n1 − t1n2 (8.13)

Here, t1 and t2 denote the surface tractions evaluated as

t1 = τ11n1 + τ12n2

t2 = τ21n1 + τ22n2

(8.14)

with the shear–stress tensor components τij defined in equation (2.60).

Making use of the expression (7.38) for the calculation of the gradientsinvolved in the computation of equations (8.12) and (8.13), we can directlydetermine the heat transfer and skin friction coefficients.


In order to compare our results with the available experimental and nu-merical data, during the simulation of the flow over a compression corner, thescaling presented by Rudy et al [14] is adopted. The plotted wall coefficientsare then given by log(50Cp), log(1000Ch) and 50Cf where the pressure, heattransfer and skin friction coefficients Cp, Ch and Cf are defined by

Cp = pw/(ρ∞U∞U∞

2)

Ch = qw/[ρ∞U∞(H∞ − Hw)]

Cf = τw/(ρ∞U2

∞

2)

(8.15)

and the subscripts w and ∞ refer to wall and free stream values respectively.

8.3.4 Further considerations

The issues of development of unstructured mesh generation and adaptivity pro-cedures suitable for viscous flow simulations is not addressed here but mustbe mentioned due to their importance. One alternative for the generation ofsuitable viscous meshes consists on the development of a mesh generator whichgenerates a structured grid in the immediate vicinity of the solid surfaces andcompletes the triangulation using the unstructured mesh approach. This pro-cedure has been successfully utilized for many realistic viscous flow simulations,see for instance [1, 12]. However, this approach is not very flexible for the in-clusion of adaptivity or the extension for general complex 3–D configurations.Alternatively, the approach, presented by Hassan et al [2] and further devel-oped in reference [3], for generating and adapting fully unstructured viscousmeshes, overcome the shortcomings mentioned above and presents as a goodchoice. Other alternatives can be devised and this area must be addressed ina future work.

8.4 Numerical Applications

Two examples involving high speed laminar viscous compressible flow are con-sidered to illustrate and compare the numerical performance of the algorithmswhich have been described. These examples involve simple geometries, whichmeans that structured triangular meshes can be adopted. This also allows fordirect comparison with the results produced by structured mesh flow solvers onclosely related grids [14, 15]. Theoretical and experimental data are also usedduring this validation study. The results presented here using the describedformulations are those given in references [7, 8].


8.4.1 Supersonic flow past a flat plate

A basic validation test for any solution algorithm for viscous flow is the com-parison of the predicted laminar boundary layer development on a flat platewith the exact solution due to van Driest [18]. The comparison will indicateif there is excessive artificial dissipation in the numerical scheme.

Figure 8.1: Adopted mesh for the supersonic flow past a flat plate problem.

The example chosen consists of a free stream flow at a Mach number of 4.0 anda temperature of 392.40R. The Reynolds number, based upon the length ofthe plate, is 4× 106. The local Prandlt number is assumed to be constant andequal to 0.75. The computational mesh employed is displayed in figure 8.1 andconsists of a grid with 20,000 elements and 10,201 nodes. This is a triangulationof a structured 101 × 101 quadrilateral grid. The mesh uses highly stretchedelements, with the streamwise spacing distribution increasing from the leadingedge (aspect ratio 1/50) to the trailing edge (aspect ratio 1/175.25). In thenormal direction, the spacing increases as we move away from the flat plate,as shown in figure 8.1. The initial conditions simulate a flat plate suddenlyexposed to the free stream. The no–slip adiabatic wall condition is imposed atthe plate. The far field boundary condition follows that described in section7.2.4 for the inviscid fluxes and that given in section 8.3.1 for the viscous fluxes.

The wall pressure distributions, computed by the G/LED and the MUSCLschemes using the Roe Riemann solver, are displayed in figure 8.2 and can be


0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0 0.2 0.4 0.6 0.8 1

P

X1

MUSCLG-LED

Figure 8.2: Supersonic flow past a flat plate. Computed pressure distributionon the plate surface.

seen to be in good agreement with each other. The MUSCL limiter, computedin an upwind-biased stencil, was adopted for the G/LED computation, whilethe Venkatakrishnan/Thomas limiter with the modifications for density andpressure described in chapter 7 is used with the MUSCL scheme.

Figure 8.3(a) shows the comparison between the similarity solution of vanDriest [18] and the velocity profiles computed using the G/LED algorithm.The velocity distribution values, normalized by the free stream velocity, aredisplayed at different sections downstream of the leading edge. The verticalscale on the figure is the dimensionless length

Y ⋆ =x2

x1

√Rex (8.16)

where Rex denotes the Reynolds number computed using the distance fromthe leading edge x.

The computed velocity profiles are seen to be in good agreement with theanalytical solution. An indication of the higher resolution achieved with theseschemes, can be obtained from figure 8.3(b), which displays the results com-puted by the first order upwind, the MUSCL and the G/LED schemes. It wasobserved that the adoption of the minimod, the MUSCL or the van Albadalimiters, with the G/LED scheme, leads to almost identical results, which arenot presented here. In figure 8.4(a), the solution obtained using the MUSCLscheme with the AUSM splitting and again the Venkatakrishnan/Thomas lim-iter is shown for different sections downstream of the leading edge. A goodprediction of the boundary layer development when compared with the simi-larity solution is obtained and the performance is similar to the MUSCL/ROE


(a) (b)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y*

U1/Uoo

X1=0.3 X1=0.5 X1=0.7 X1=0.9 van Driest

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y*

U1/Uoo

1th-Upw MUSCL

G-LED van Driest

Figure 8.3: Supersonic flow past a flat plate. Computed velocity profiles on(a) G/LED results at different cords and (b) results comparison at x1 = 0.5.

results of figure 8.3(b). Only small differences were observed when differentsupports, symmetric or upwind, were used for the computation of the lim-iters, with the G/LED formulation. The slightly different results can be seenin figure 8.4(b), where the minimod limiter is adopted. For this application,the influence of the use of different supports for the limiter calculation proveto be not a very important issue. As a result, we can say that the artificialdissipation induced by these limiters is basically the same, in this simple testcase, and both MUSCL and G/LED schemes succeed in predicting the laminarboundary layer, presenting good agreement with the theoretical solution.

8.4.2 Hypersonic flow over a compression corner

The second example consists of a hypersonic flow over a compression corner of240 angle. The free stream flow is at a Mach number of 14.1 and the Reynoldsnumber is 103, 680, based upon a flat plate length of 1.44 ft. The temperatureof the fluid in the free stream is 1600R and the local Prandlt number is constant


(a) (b)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

Y*

U1/Uoo

X=0.5 X=0.7 X=0.9

van Driest

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y*

U1/Uoo

LED-Upw LED-Sym van Driest

Figure 8.4: Supersonic flow past a flat plate. Computed velocity profiles. (a)Using the MUSCL/AUSM formulation and (b) Using G/LED Scheme withdifferent supports for limiter calculation.

at the value 0.72. The temperature of the wall is held fixed at 5350R. TheReynolds number is low enough to ensure that the flow remains completelylaminar and the free stream temperature is low enough so that there are nosignificant real-gas effects [14]. A schematic description of the expected flowbehaviour is shown in figure 8.5.

The resolution of the leading edge shock, of the substantially thin bound-ary layer on the ramp, of the separation zone and of the interaction shock–shock and shock–boundary layer, etc are very demanding, making this problema challenging application in the validation of any numerical algorithm for thesolution of the Navier–Stokes equations. This problem has been extensivelystudied in the literature and both numerical [15] and experimental data [4] areavailable for comparison.

Computations were made using a triangulation of a 111 by 101 structuredgrid (figure 8.6) using both G/LED and MUSCL schemes. The domain ofcomputation was extended ahead of the leading edge of the flat plate. This


Flow

Leading Edge

Shock

Compression

Fan

Induced

ResultantShock

Slip Surface

Expansion Fan

RecirculationPoint

Separation

ReattachmentPoint

BoundaryLayer

Shock

Figure 8.5: Schematic solution behavior for the hypersonic flow over a com-pression corner of 240 angle test case.

extension was found to be necessary to remove non–physical behavior from thesolution in the region near the leading edge of the plate, i.e. only a very smallpressure rise on the leading edge and the presence of some oscillations wasfound for any scheme tested. The no slip isothermal condition was imposedat the wall. The far field boundary condition follows that described in section7.2.4 for the inviscid fluxes and that given in section 8.3.1 for the viscous fluxes.

Figure 8.6: Adopted mesh for the hypersonic flow over a compression cornerof 240 angle simulation.

For this problem, the minimod, MUSCL or van Albada limiters were allemployed with the G/LED scheme. It was detected that the adoption of anupwind–biased stencil for the computation of the limiters leads to less artificialdiffusion, confirming previous observations when simulating the compressible


Euler equations [9]. Furthermore, it was found that the results produced wereclosest to those obtained by other methods when the van Albada limiter wasused. It should be mentioned that, the adoption of mixing the limiters and/orsupport for their calculations, when using characteristic variables, was foundto have a big influence in the solution in the region downstream of the reattach-ment point. The very thin boundary layer and the existence of an expansionfan might explain such influence of different limiters and support adopted intheir calculations. Apart from this region, the solutions were very similar tothat obtained with the van Albada limiter. The adoption of symmetric lim-iters with the G/LED scheme produces more diffusive shock representationsand overall worse results than the results obtained using upwind–biased lim-iters.

Figure 8.7 shows the density, pressure coefficient and Mach number con-tours for the G/LED scheme with the van Albada limiter, computed in anupwind–biased stencil. The corresponding plots for the MUSCL scheme arebasically the same and are not shown here. Qualitative aspects of the flow field,such as the presence of a separation zone and the form of the shock–boundarylayer interaction, are well represented.

The computed velocity vectors are presented in figure 8.8, where most ofthe features shown in figure 8.5 are apparent.

Rudy et al [14, 15] have presented wall plots computed for this problemusing the structured mesh flow solver CFL3D. The plotted wall coefficientsare computed according to the definitions given in equation (8.15). In Fig-ure 8.9(a), the predicted variation of the pressure coefficient along the wall,for both the G/LED and the MUSCL schemes, using the Roe Riemann solver,is compared with the results of the CFL3D code. Apart from the region veryclose to the separation point and at the pressure peak, the three solutions arevery similar. The results of the MUSCL scheme appear to be the most smearedand the results of CFL3D are the sharpest near the separation point. The com-parison of the predicted variation of the heat transfer coefficient along the wallfor the three schemes is given in figure 8.9(b). Again, CFL3D gives sharperresolution near the separation point and slightly different values in the separa-tion zone, where the MUSCL and G/LED predictions are very close. Similardifferences through the separation zone is present in the grid–converged resultsshown in reference [3]. Figure 8.9(c) shows the predicted variation of the skinfriction coefficient along the wall. In this case, the main discrepancy betweenthe present schemes and CFL3D occurs in the region downstream of the reat-tachment point. The difference is small but is present throughout the rampand the predicted peak is higher here than with the CFL3D algorithm. A skinfriction prediction which are closed to the CFL3D results is produced if theG/LED method and the upwind–biased MUSCL limiter is adopted. These


Density

Pressure Coefficient

Mach Number

Figure 8.7: Hypersonic flow over a compression corner of 240 angle. Computeddistribution of density, pressure coefficient and Mach number contours.


Figure 8.8: Hypersonic flow over a compression corner of 240 angle. Computedvelocity vectors distribution


(a) Pressure Coefficient

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cp

X1

Exper. G-LED

MUSCL-ROECFL3D

(b) Heat Transfer Coefficient

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Ch

X1

Exper. G-LED

MUSCL-ROECFL3D

(c) Skin Friction Coefficient

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cf

X1

Exper. G-LED

MUSCL-ROECFL3D

Figure 8.9: Hypersonic flow over a compression corner of 240 angle. Compar-ison of Computed Surface Distributions.


results are not so close to the CFL3D results over the rest of the wall and arenot shown here.

The results of experimental measurements have also been included in thesethree figures. It is apparent that the comparison between numerical predictionand experiment is not good and the present computations tend to support theclaim [15] that significant 3D effects were present in the original experiment.

In figure 8.10, the predicted variation of the pressure coefficient, heattransfer coefficient and skin friction coefficient along the wall, computed usingthe MUSCL/AUSM scheme are compared again with the results of the CFL3Dcode. It was found that the choice of a split–Mach version (see discussionin section 5.4.3) is crucial when the Navier–Stokes equations are considered,and the use of a split–velocity version lead to meaningless wall coefficients,despite a reasonably good–looking overall solution, in terms of contours. It wasalso observed that the higher–order AUSM scheme gives the sharpest shockresolution, with the best results when compared to the predictions obtained byCFL3D. The differences through the separation zone persists and the resultswith the MUSCL/AUSM scheme is closer to that given with the other analysedschemes.

Evidence from other researchers shows that accurate prediction of thiscomplex flow field depends strongly on a good resolution of the shock wavewhich emanates from the leading edge of the plate. This might explain thebetter performance of the AUSM basis MUSCL scheme, as very good shockcapture was achieved for 1–D simulations even using a first–order scheme.A very small amount of artificial dissipation is added, which allows a sharpcapture of the separation. Furthermore, the proper upwinding of the con-vective terms and pressure splitting inherent in this scheme justify its goodperformance for high speed simulations [5]. However, some oscillations on thepredicted pressure, at the first points on the flat plate, can be noted in figure8.10(a).

A severe discontinuity is present at the leading edge with the Mach numberdropping from 14.1 to 0.0 in a region where the physical viscosity is still prac-tically zero. This demanding strong shock requires proper amount of artificialdissipation in order to damp oscillations without compromising the captureof the discontinuity. A zoom on the pressure distribution obtained for theschemes which adopt the Roe Riemann solver shows some lightly oscillationstoo, which are not noticeable in the plotting scale of the figure 8.9. A close in-vestigation on the velocity vectors shows that for the first few layers of points,in the streamwise direction, the leading edge shock lies between the wall andthe first layer of points in the normal direction. This indicates the necessityof a higher mesh density in the vertical direction to resolve properly the shockin the neighbourhood of the leading edge.


(a) Pressure Coefficient

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cp

X1

Exper. MUSCL-AUSM

CFL3D

(b) Heat Transfer Coefficient

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Ch

X1

Exper. MUSCL-AUSM

CFL3D

(c) Skin Friction Coefficient

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cf

X1

Exper. MUSCL-AUSM

CFL3D

Figure 8.10: Hypersonic flow over a compression corner of 240 angle. Com-parison of Computed Surface Distributions.


However, despite the expected improvement in the solution with the use ofa suitable mesh refinement strategy, it might not, ultimately, eliminate thepresence of oscillations when using the AUSM flux splitting.

The explanation for the wiggles in the 1–D simulation of the hypersoniccolliding flow when using the hybrid FD/FV splitting approach (see section5.6.2) is believed to be the main reason for the present bad behavior. Thisis further supported by the fact that first–order AUSM solution also presentthe oscillations at the vicinity of the leading edge. The results obtained inthe first attempt to use the modified AUSMDV and AUSMDV formulationsof Wada and Liou [19] eliminate the oscillations but were disastrous, with thesolutions resembling first–order predictions. The close connection between theAUSMV and the Van Leer flux vector splitting justify the bad results obtainedin the AUSMV computation. The free parameter (see [19]), which regulates theFD or FV nature of the AUSMDV scheme, has an important role for viscoussimulations. The incorrect choice of the value for this parameter might be thereason for the bad results obtained using the AUSMDV approach. Furtherinvestigation is required for a definite conclusion on this issue.


We have shown in this chapter how techniques developed originally for 1–D flowsolvers, later extended for 2–D Euler calculations, can be further extended toproduce high–order accuracy procedures for the solution of the 2–D compress-ible Navier-Stokes equations on unstructured triangular grids. The utilizationof a rational way of supplying additional numerical dissipation by means oflimiting procedures make the present schemes robust and this characteristic isvery important when complex flow simulations are attempted.

Although the influence of limiters on accuracy, stability and convergencerate has been shown to be important for the solution of the Euler equations, ingeneral the use of most limiters tend to produce reasonable results for inviscidflows as shown in chapter 7. Nevertheless, when the Navier–Stokes equationsare considered, especially at high Reynolds number and in the hypersonicregime, the choice of limiter was found to be crucial to ensure that the nu-merical dissipation induced does not exceed the physical dissipation. Upwind–biased limiters, which are computed using a more physical and smaller support,tend in general to give more accurate results. Here, of the limiters analysedfor the G/LED scheme, the limiter of van Albada supplied the best resultswhen compared to other numerical schemes. However, further investigation isrequired before a definite conclusion can be drawn about which limiter shouldbe preferred in general. Moreover, any improvement in accuracy is frequentlyfound to be accompanied by a deterioration in convergence rate and stability


behaviour. In such cases, the adoption of a strategy involving the freezingof the limiters discussed in chapter 7, was found to help convergence of theresidual towards machine zero. The effect of adding background dissipation,e.g. LW/LED or SLIP(2), which should improve convergence behavior, was notanalysed for viscous applications.

For viscous computations, the split–velocity option for the AUSM schemeproduces oscillations in the solution in the vicinity of viscous walls. For theseproblems, therefore, the use of the split–Mach version is essential for a mean-ingful result.

In general, a good correlation was found between the predictions of thedifferent high–resolution schemes considered in this thesis. The results demon-strate that all analysed schemes, accurately predict complex flow configura-tions which incorporate viscous–inviscid interactions. No emphasis on theissue of relative efficiency of each scheme was addressed in the present study.Although a bad behaviour for the pressure distribution in the vicinity of theleading edge was achieved, the overall numerical results obtained for the ap-plications analysed, the efficiency in terms of CPU time and also the favorableproperties of the AUSM formulation make this scheme particularly promis-ing for viscous hypersonic complex flow computations, and in particular forblunt–body flow simulations.

The necessity to concentrate a large number of points close to solid wallsin order to resolve the boundary layer with sufficient accuracy, mainly at highReynolds number, requires highly stretched elements. Apart from the big re-duction on the allowable time–step, with severe implications on convergencetowards a steady–state solution using an explicit time integration, the viscouslayer meshes very frequently lead to aspect rations bigger than 1/200. The pos-sible adverse effect in terms of accuracy when adopting these highly stretchedelements were not carefully analysed here. However, it can be said that for thepresent computations, the analysed schemes present good behavior using theorder of stretching mentioned above.

The proposed schemes may be employed with general triangular grids andthis flexibility should be exploited in subsequent works with the use of adaptivemesh refinement to enhance the accuracy of the computed solution. Applica-tions such as viscous hypersonic flows over blunt bodies and time dependentflows will have to be successfully addressed in order to prove the general va-lidity of the presented approaches.

Bibliography

[1] O. HASSAN. Finite Element Computation of High Speed Viscous Com-pressible Flows. PhD thesis, University College of Swansea, 1990.

[2] O. HASSAN, K. MORGAN, J. PERAIRE, E.J. PROBERT, and R.R.THAREJA. Adaptive Unstructured Mesh Methods for Steady ViscousFlow. Technical Report 91–1538, AIAA Paper, 1991.


[4] M.S. HOLDEN and J.R. MOSSELE. Theoretical and Experimental Stud-ies of the Shock Wave–Boundary Layer Interaction on Compression Sur-faces in Hypersonic Flow. Technical Report ARL 70–0002, CALSPANReport, University of Bufalo Research Center, 1970.


[6] H. LUO, J.D. BAUM, R. LOHNER, and J. CABELLO. Adaptive Edge–Based Finite Element Schemes for the Euler and Navier–Stokes Equationson Unstructured Grids. Technical Report 93–0336, AIAA Paper, 1993.

[7] P.R.M. LYRA, M.T. MANZARI, K. MORGAN, O. HASSAN, andJ. PERAIRE. Side–Based Unstructured Grid Algorithms for Compress-ible Viscous Flow Computations. Technical report, University College ofSwansea Report CR/817/94, 1994. Also accepted for publication in theInt. J. for Engng. Analysis and Design.

[8] P.R.M. LYRA, K. MORGAN, and J. PERAIRE. A High–ResolutionFlux Splitting Scheme for the Solution of the Compressible Navier–StokesEquations on Triangular Grids. In Proc. of the International Workshopon Numerical Methods for Navier–Stokes Equations, Heidelberg, 1994.

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Vieweg publishers, in press. Also published internally at University Col-lege of Swansea Report CR/818/94 (1994).


[10] M.T. MANZARI. Private Communication, January 1994.


[12] J. PERAIRE, K. MORGAN, and J. PEIRO. Unstructured Mesh Methodsfor CFD. Technical Report 90–04, I.C. Aero Report, 1990.


[14] D.H. RUDY, J.L. THOMAS, A. KUMAR, P.A. GNOFFO, and S.R.CHAKRAVARTHY. A Validation Study of Four Navier–Stokes Codesfor High–Speed Flows. Technical Report 89–1838, AIAA Paper, 1989.

[15] D.H. RUDY, J.L. THOMAS, A. KUMAR, P.A. GNOFFO, and S.R.CHAKRAVARTHY. Computation of Laminar Hypersonic Compression–Corner Flows. AIAA Journal, 29:1108–1113, 1991.

[16] S. SOLTANI. An Upwind Scheme for the Equations of Compressible Flowon Unstructured Grids. PhD thesis, University of London – ImperialCollege of Science, Technology and Medicine, 1991.

[17] R.C. SWANSON, E. TURKEL, and J.A. WHITE. An Effective MultigridMethod for High–Speed Flows. In Proc. of the Fifth Copper MountainConference on Multigrid Methods, 1991.

[18] E.R. VAN DRIEST. Investigation of Laminar Boundary Layer in Com-pressible Fluids Using the Crocco Method. Technical Report 2597, NASATechnical Note, 1952.

[19] Y. WADA and M.-L. LIOU. A Flux Splitting Scheme with High–Resolution and Robustness for Discontinuities. Technical Report 93–0083,AIAA Paper, 1994.

Chapter 9

Conclusions

The endeavor to write about the research work developed during my stay inSwansea in a somewhat self–contained thesis is a result of the excitement andchallenge that I have experienced in working in the field of computational fluiddynamics. I hope that sufficient background information has been providedto allow the reader to follow and evaluate the proposed formulations and theresults of the present study. I also hope that this thesis will supply elementsfor further investigation on the specific topics discussed here.

Relevant conclusions, remarks and the author’s experience and opinion tospecific topics as well as suggestions for extensions of this work have been pre-sented from time to time throughout this thesis. Therefore, this thesis comesto an end with a briefly summary of just the main achievements accomplished,by drawing few general conclusions and finally by presenting some suggestionsfor useful extensions of the present work and areas for future studies.

9.1 Summary of Achievements

In the present work, computational codes have been developed for steady–state and transient simulation of fluid dynamics and heat conduction prob-lems. Each of these computational tools has many ingredients concerning themain requirements (reliability, robustness, efficiency and versatility) necessaryfor the success of any designed numerical procedure to deal with engineeringand scientific applications. Apart from some specific elements of each of theparticular formulations adopted, some of the common features embodied in allthree distinct research works pursued here includes:

• The use of the finite element method as the basis for the spatial discreti-sation;

Conclusions 336

• Utilization of data structures suitable for general domain discretisation;

• Incorporation of procedures to assess the accuracy of the solution;

• Possibility for mesh adaptivity;

• The adoption of some form of automatic time–step control;

• Adoption of formulations with reduced number of free–parameters;

• The use of techniques to enhance stability behaviour of explicit timeintegration and/or the possibility for the adoption of implicitly time–stepping procedure;

• Incorporation of some techniques to improve efficiency of the computa-tions.

In general, the modern shock–capturing schemes analysed here presentgood performance for applications which contain moderate to strong discon-tinuities. The proposed procedure to extend to multi–dimensions and to vis-cous simulations, via a Galerkin edge–based finite element procedure, includingmany features described for enhancement of the formulations, is robust enoughto deal with a wide range of compressible fluid flow applications. When deal-ing with problems with no discontinuities, the overall accuracy of the schemescan be degraded, in some instances, since most schemes reduce to first–orderin the vicinity of extrema. Bad performance of shock–capturing schemes isoften experienced when they are directly applied for nearly incompressible orvery low Mach number flows [12], as the acoustic wave speed goes to infinity,resulting in an ill–conditioned problem. The methods suitable for compress-ible equations must be adapted, normally using some variant of the pseudo–compressibility method introduced by Chorin [2, 4, 8]. Another option, is touse specific methods specially developed for incompressible applications, suchas the least–squares Petrov–Galerkin formulation described in this work. Thisformulation circumvents stability problems associated with incompressibilityand introduces an automatic correct amount of streamline upwinding. Whensimulating the scalar transient heat conduction equation, the adoption of aGalerkin finite element formulation combined with a fully implicit time inte-gration represents a very robust basic formulation to deal with practical heatconduction applications. All the procedures described here are capable of solv-ing a very wide range of practical engineering problems and are extendible forboth the analysis of 3–D problem and the modeling of more complex phys-ical phenomena. The implementation of the computer codes developed haselements which make them feasible for implementation on vector processorcomputer configurations, as already accomplished for the compressible fluidflow computational tools described here.

Conclusions 337

Instead of presenting a list of the specific achievements in each area stud-ied I decide to present a couple of interesting problems faced through thedevelopment of each of the three stages of this research work and the solutionsproposed for them.

The development of a simultaneous adaptive strategy, which encompassesmesh refinement and automatic time–step control, results in some numericalimplications. The most important implication is the fact that the error in thetemporal and spatial discretisations are related to each other, independent ofthe discretisation techniques adopted. One interesting issue of the combinedadaptive strategy is the necessity to impose a lower–bound on the time–stepsize to avoid oscillation in the solution, when the standard consistent massmatrix is adopted in finite element formulations. This fact is, in general, notknown by finite element practitioners.

The presence of singularities and other features which dominate the dis-crete error in the approximate solution can either damage the effectivenessof the adaptive refinement procedure, based on the principle of error equidis-tribution, or require the introduction of arbitrary parameters to control therefinement procedure. The use of a two–step strategy for the mesh refinementpresents as an alternative, which was extended here to deal with the remesh-ing adaptive procedure. This strategy allows for fully automatic adaptivitywithout the necessity of arbitrary parameters, even when distinct features andsingularities are present in the solution.

The detection of pressure oscillations in the incompressible fluid flow solu-tion when the pressure–continuity equation is integrated using a global time–step, using the described least–squares Petrov–Galerkin formulation, was notobserved before the implementation of the adaptive mesh refinement proce-dure. A priori, solely uniform meshes were adopted and this problem washidden. The adoption of a local time–step, which in the presented formula-tion is related to the upwind–optimal parameter, corrects this problem and ispossible due to the elliptic nature of the pressure–continuity equation.

The hitherto unnoticed adverse side effect of the repeated interpolationbetween meshes inherent in an unsteady remeshing adaptive procedure, whenfirst–order interpolation is used, was observed in the numerical applicationsperformed here. It was shown that the use of first–order interpolation acts asa strong second–order dissipation, which compromises the overall performanceof the results. One possible way to overcome such a problem by means of aselective higher–order interpolation was described and successfully validated.

When designing the local 1–D stencil, required for the extension of the1–D high–resolution shock–capturing schemes, we faced the problem of lackof robustness, of the resultant limited schemes, for hypersonic calculationswhen using the gradient reconstructions described. The use of linear interpo-lation considering the adjacent triangle to the edge under consideration also

Conclusions 338

suffer from similar weakness, for some applications attempted. This led to theadoption of the actual triangle which contains the dummy node to furnish therequired information for the linear interpolation. This option proves robustfor all the applications performed. Despite the impossibility of proving theLED property for the resultant schemes, which uses this stencil construction,oscillation free results were obtained.

The possibility to have local spurious negative values for the thermody-namic variables in different stages of the solution algorithms, when severe flowconditions are examined, requires specific strategies to either prevent or cir-cumvent such a problem. The incorporation of a relaxation during the updat-ing procedure, the modification of certain limiter functions and the eliminationof dummy edges, whenever density or pressure would become negative, pre-vents failure of the procedure.

The inability of the AUSM flux splitting approach, with the split–velocityversion as the interface convective quantity, when dealing with viscous simula-tions, despite the good results obtained for the inviscid calculations, representsanother unexpected problem, not mentioned in the literature, related to thisscheme. The use of the split–Mach alternative repairs this problem. Thepresence of oscillations near the stagnation point for supersonic viscous flowcomputation using AUSM remains unsolved, but a possible explanation wasgiven and the remedy is still under investigation.

These are just few problems that I came across here and they give an ideaof the sometimes arduous, but at the same time rewarding, process of scientificinvestigation. Many other issues, either generic to any numerical procedure orspecific to each class of applications attempted in this work, were addressedthroughout the thesis and are not repeated here.

To sum up, the resultant computational tools represent the primary ac-complishment of this work. The explanation for the reason behind the prob-lems faced in the course of this work and the remedies proposed for themrepresent, in my opinion, the most important conceptual contributions of thepresent work. The use of a unified notation and description of distinct formu-lations in the same framework, particularly when dealing with compressiblefluid flow, allows better understanding and comparison of these formulations.The discrete formulations, data–structure and various strategies adopted aresuch that they can be directly extended for 3–D analysis. In addition, theylead to codes with reduced memory requirements and, when using edge–baseddata structure, also reduced CPU time when compared to codes which use thetraditional element based data structure [7, 9]. Finally, the assessment of theperformance of the various procedures, during the validation study, furnisheselements which are helpful to reduce uncertainties and to give an insight intoelements and parameters existent in each procedure.

Conclusions 339

9.2 General Conclusions

In this work, the author tried to emphasize the importance that 1–D mod-els performed and still exercise on the development of theories and numericalschemes in CFD. Apart from that, the 1–D models represent a remarkablyeffective and instructive “numerical laboratory”, allowing the fluid dynamicistto acquaint himself with many peculiarities of numerical simulation and totest, compare and perform an initial validation of new theories. Of course,the maxim that a failure of a scheme to deal with 1–D model will normallyrepresent a failure with the higher–dimensional extensions, while the successwith a 1–D model doesn’t guarantee success with the correspondent extensionpersists. Intensive experimentation on the desired spatial level of approxima-tion must be pursued for the validation of the scheme. Furthermore, issuessuch as boundary conditions, adaptivity, robustness, computational efficiency,etc have little sense in 1–D calculations.

The use of well grounded mathematical theories and the mathematical rat-ification of some methodologies developed purely from necessity and intuitionis of paramount importance to allow methods to flourish. However, it shouldnot represent a barrier, but an auxiliary tool, during the innovative stage ofdeveloping practical computational codes. This allows engineers and appliedscientists to tackle problems and develop tools for which solid mathematicalbackground are still to be presented. However, this procedure normally in-troduces uncertainties and problem adjustable parameters. Unfortunately, noscheme analysed, or even in existence to date, can really be considered freefrom adjustable parameters or elements to be tuned and free from trial anderror procedure. In addition, even in science, we have preferences and theanalysis of some results or performance of different schemes are not free fromfavouritism. In this way, I can say that the best scheme for a certain classof problems, after passing a scientific and practical validation stage, is muchmore a question of taste and familiarity than anything else. Finally, I wouldsay that the myth of “the ultimate scheme” is likely to persist, maybe for ever,and the experience of the fluid dynamicist will always be needful to deal withthe uncertainty and numerous elements of CFD.

9.3 Suggestions for Further Research

Further and future works can be carried out in many distinct areas. A sum-mary of some of the most important issues for the near future in CFD can beextracted from the words of Van Leer [11], which state that: CFD algorithmsfor the coming generation of massively parallel computers will have to be ex-tremely robust. They will most likely be implemented on adaptive unstructured

Conclusions 340

grids, and will be used for ambitious steady and unsteady three–dimensionalflows. In such a complex environment there is little place left for hand–tuningparameters that regulate accuracy, stability and convergence of the computa-tions. Such algorithms already exist in CFD, but a lot still has to be done inany of the above mentioned areas of research.

Reducing the number of free–parameters – Although the dissipation providedby upwinding is in general enough to stabilize and to give oscillation–free phys-ically meaningful solutions, any existing scheme based upon a flux–splittingapproach is not free from imperfections, with quite amazing failures in someinstances [10]. This is especially true when dealing with multi–dimensionalextensions of the 1–D type schemes. These shortcomings can normally becontrolled by a judicious addition of artificial dissipation in the same fashionas the entropy fix, and successful simulations have been possible. However,one should be aware of this issue, of the necessary lost of accuracy inherentin such strategies and of the extra approximations involved during the multi–dimensional extension. The use of genuinely multi–dimensional upwinding, ei-ther through further developing the Generalized Galerkin finite element basedmethods [1, 5, 6, 7] or through multi–dimensional differencing schemes meth-ods [3, 11], which are not yet fully mature, represent one alternative whichmight overcome many of the deficiencies of the current approaches. The devel-opment of such a methods presents a great challenging area of research for theforthcoming years. Any progress in such areas will, it is to be hoped, reducethe necessity for free–parameters in the resultant algorithm.

Tackling ambitious simulations – The computation of complex viscous flows,possibly in the turbulent regime, considering real gases effects, chemical reac-tion, electromagnetic effects coupled, etc, still represents a tremendous chal-lenge for CFD practitioners. Not only efficient numerical techniques must bedeveloped but adequate models are still under investigation. Experiments ofhigh standard level of accuracy must be provided for the validation of any pro-posed scheme. Some of these problems require also a multi–disciplinary effortto be undertaken.

Modelling three–dimensional configurations – The impressive progress in com-puter technology and the availability of supercomputers has attracted the at-tention of the scientific community towards three-dimensional complex flowsimulation. Despite the fact that the objective of “calculation over a completeaircraft configuration” has, to a certain extent, been achieved, this area stillrepresents a great challenge. Apart from some specific problems one might, andalmost certainly will, face when extending a two–dimensional formulation intothree-dimension, the main issues here are related to the development of efficientmesh generators, visualization tools and to the design of data–structures andtechniques which allow savings in terms of memory and CPU requirements.Concentrated effort on the development of multigrid acceleration techniques,

Conclusions 341

adaptive implicit or implicit/explicit time integration strategies are some ofthe areas which still need further improvements, mainly when considering theunstructured grid philosophy.

Exploiting the modern computer configurations – Finally, it doesn’t matter howclever a specific algorithm is, the longevity and industrial interest in the algo-rithm is likely to depend on how efficient it can perform on the new generationof computer architectures to allow impact on design. The research centredon how the current algorithms can exploit parallel computer architectures andhow related implementations issues can be tackled are unavoidable, otherwisewe are misusing an available resource and being at least economically wasteful.

Bibliography

[1] J.C. CARETTE, H. DECONINCK, H. PAILLERE, and P.L. ROE. Mul-tidimensional Upwinding: Its Relation to Finite Elements. In Proc. ofVIII International Conference on Finite Elements in Fluids: New Trendsand Applications, pages 446–456, 1993.

[2] A.J. CHORIN. A numerical Method for Solving Incompressible ViscousProblems. J. Comp. Phys., 2:12–26, 1967.

[3] H. DECONINCK, R. STRUIJS, G. BOURGOIS, H. PAILLERE, and P.L.ROE. Multidimensional Upwind Methods for Unstructured Grids. InAGARD Report 787 on Special Course on Unstructured Grid Methods forAdvection Dominated Flows, pages 4.1–5.17, 1992.


[5] T.J.R. HUGHES and M. MALLET. A New Finite Element Formulationfor Computational Fluid Dynamics: IV. A Discontinuity Capturing Op-erator for Multidimensional Advection–Diffusive Systems. Comp. Meth.Appl. Mech. Engng., 58:329–336, 1986.

[6] C. JOHNSON, A. SZEPESSY, and P. HANSBO. On the Convergence ofShock–Capturing Streamline Diffusion Finite Element Methods for Hy-perbolic Conservation Laws. Math. Comp., 54:82–107, 1990.


[8] B. MULLER, J. SESTERHENN, and H. THOMANN. Preconditioningand Flux Vector Splitting for Compressible Low Mach Number Flow. InLecture Notes on Physiscs 414 – 13th International Conference on Numer-ical Methods in Fluid Dynamics, pages 125–129, Rome, 1992. Springer–Verlag.

342

Conclusions 343


[10] J.J. QUIRK. A Contribution to the Great Riemann Solver Debate. Tech-nical Report 92–64, ICASE Report, 1992.

[11] B. VAN LEER. Progress in Multidimensional Upwind Differencing. InLecture Notes on Physiscs 414 – 13th International Conference on Nu-merical Methods in Fluid Dynamics, pages 1–26, Rome, 1992. Springer–Verlag.


Appendix A

Time step lower limit

In this appendix a summary of the demonstration given by Rank et al [3] ofthe existence of a lower bound for the time–step size when the finite elementformulation is adopted for the spatial discretisation of a linear isotropic heattransfer problem (3.1), is presented. Assume ρ, Cp, k are constants and Q, Tare given functions, which are assumed sufficiently smooth [1]. Following theprocedure described in section 3.2, the algebraic system of equations given inequations (3.10) and (3.11) is rewritten here for the Euler–backward scheme(θ = 1) as

K∗Tn+1 = F∗

T = T 0 at t = t0(A.1)

where

K∗ = K +ρCp

∆tn+1M (A.2)

with the independent vector F∗

and the conductivity matrix K determined aspresented in section 3.2. The capacity matrix C is written here as (ρCp)×M

and

[M]IJ =∑

E∈I

[ ∫

ΩE

NINJ dΩ]

(A.3)

for each I = 1, . . . , p , where the summations just extend over those elementsE which contain node I, and p is number of discrete points. As a consequenceof the ellipcity of the conduction term, and if the discretisation satisfies theregularity condition [3], the following relations are valid

Time step lower limit 345

Kij ≤ 0 ∀ i 6= j and Mij ≥ 0 ∀ i, j (A.4)

Ciarlet and Raviart [1] proved that the discrete analog of the propertyknown as the maximum principle, which guarantees stability and uniform con-vergence in an elliptic problem, is the monotone character of the matrix K. Amatrix K is called monotone if it is invertible and its inverse is non–negative,i.e. (Kij)

−1 ≥ 0 ∀ i, j. They also prove that if K is diagonally dominant andcondition (A.4) holds then K is monotone. For each time–level, the parabolictransient heat problem reduces to an elliptic problem and the matrix K∗ mustpossess the monotone character in order to satisfy the discrete maximum prin-ciple. As ρCp ≥ 0, for a given finite element discretisation and conductivity Kthere is a limit

ρCp

∆tn+1≤ L or ∆tn+1 ≥ ρCp

L= ∆tlb (A.5)

such that

K∗ij = Kij +

ρCp

∆tn+1Mij ≤ 0 ∀ i 6= j (A.6)

and ∆tlb represents a lower bound limit for the time–step magnitude necessaryto guarantee the monotone character for K∗. For a one–dimensional modelproblem, with uniform discretisation, the expression (A.5), according to Ranket al [3], reduces to

∆tn+1 ≥ ρCp

6K(∆x)2 = ∆tlb (A.7)

Using a parametric study, an extension of this limit for two–dimensionis presented by Murti et al [2], who proposed, for isoparametric elements thelimit given by

∆tn+1 ≥ ζρCpH2min

3K= ∆tlb (A.8)

Here ζ is a correction factor, introduced to take into account other featuressuch as the nature of the thermal loads, type of element, aspect ration of thediscretisation, etc. In (A.8) Hmin denotes the minimum spacing in the grid.Murti et al [2] found ζ = 2.0 for a bi–linear isoparametric element and anirregular grid, which agreed with our own experiments for the model problemspresented in chapter 3.

In a stiff problem, the accuracy requirements normally impose a time–step size that we cannot afford to choose a spatial discretisation which is


compatible with the condition (A.8). However, if the matrix M is replacedby a “positive” lumped or diagonal matrix ML, then the matrix K∗ will bediagonally dominant and off–diagonally negative. This means that the (A.6)is always true and monotone convergence is assured. The use of a lumpedmatrix is recommended for this class of problems despite the lost of accuracyin the spatial discretisation associated with such a choice.

Bibliography

[1] P.G. CIARLET and R.A. RAVIART. Maximum Principle and UniformConvergence for Finite Element Method. Comp. Methods Appl. Mech.Engng., 2:17–31, 1973.

[2] V. MURTI, S. VALLIAPPAN, and KHALILI-NAGHADEH. Time StepConstraints in Finite Element Analysis of Poisson Type Equation. Comp.Methods Appl. Mech. Engng., 31:269–273, 1989.

[3] E. RANK, C. KATZ, and H. WERNER. On the Importance of the DiscreteMaximum Principle in Transient Analysis Using Finite Element Method.Int. J. Num. Meth. Engng., 19:1771–1782, 1983.

347

Appendix B

Evaluation of | A | · Z

In this Appendix, the computationally efficient procedure derived by Turkel [1]to perform the product of the absolute value of the inviscid Jacobian matrix|A| times a generic vector Z for two–dimensions, is described. Such prod-uct appears in certain first–order upwind schemes and in the matrix artificialdiffusion central–type schemes.

We define the velocity vector v, the unit direction vector n in which |A|should be evaluated and the component un of v in this direction according to

v = (u1, u2), n = (n1, n2), un = ujnj with j = 1, 2 (B.1)

The eigenvalues of |A| in direction n are

λ1(U) = un + c, λ2(U) = un − c, λ3(U) = un, λ4(U) = un (B.2)

In practical implementation the λk are normally modified to prevent λk fromapproaching zero and a possible way is described in equation (5.85). Consid-ering

σ1 =|λ1| + |λ2|

2, σ2 =

|λ1| − |λ2|2

, σ3 = |λ3| (B.3)

and defining the auxiliary vectors

L = (γ − 1)

[(u2

1 + u22)

2,−u1,−u2, 1

]M = [−un, n1, n2, 0]

R = [1, u1, u2, H ] S = [0, n1, n2, un]

(B.4)

Evaluation of | A | · Z 350

The elements of the product D = |A|Z are computed according to

di = σ3zi +[(σ1 − σ3

c2)ljzj +

σ2

cmjzj

]ri +

[σ2

cljzj + (σ1 − σ3)mjzj

]si (B.5)

for i, j = 1, . . . , 4.

When a 1–D model is adopted the following simplifications apply: i, j =1, . . . , 3, v = u1, n = n1 = 1, un = u1, with the third term of vectors L, M,R and S vanishing.

Bibliography

[1] E. TURKEL. Improving the Accuracy of Central Difference Schemes. Tech-nical Report 88–53, ICASE Report, 1988.

351

Appendix C

Finite Differences InterpretedUsing Characteristic andPolynomial InterpolationTheory

In this appendix a very elucidative analysis of the finite difference approachto solve a linear scalar hyperbolic equation is presented. This analysis [4]consists in the interpretation of finite difference schemes as a result of tracingback the characteristic from point (xI , t

n+1) to find its intersection at time tnand performing a polynomial interpolation using the available solution to findthe value of the variable on that characteristic.

Consider the 1–D advection equation

∂u

∂t+ a

∂u

∂x= 0 (C.1)

with a > 0. The solution at point I at time–level tn+1 can be computed usingthe characteristic that passes through the point (xI , t

n+1) in an x–t plane, aspresented in figure C.1.

As discussed in section 5.2.1, the solution u(x, t) is constant along thecharacteristic and then, from equation (5.4), we have

un+1I = u(xI , t

n+1) = u(x − a∆t, tn) = unk (C.2)

It is then necessary to compute unk to obtain the desired solution un+1

I . Thiscan be done by using Lagrange interpolating polynomials [6]. The Lagrangeinterpolation can be conveniently defined for a general P–order polynomialrepresentation of u according to

FD as Characteristic Tracing and Solution Interpolation 353

t

∆x

∆ ta x

n+1

n

I

I+1I-1

∆ t

I+2I-2

K

characteristic

Figure C.1: Characteristic line through (xI , tn+1) for the scalar linear advection

equation, with a > 0.

uP (xk) =I0+P∑

i=I0,∆I

Li(xk)u(xi) with Li(xk) =I0+P∏

j=I0,∆I

j 6=i

(xk − xj)

(xi − xj)(C.3)

The first–order upwind scheme [1] evaluates unk using the two nearest

points, in the upstream direction, for a linear interpolation (P = 1). Makinguse of (C.2) and (C.3), with I0 = I − 1 and ∆I = 1, one gets

un+1I = un

k =(

a∆t

∆x

)un

I−1 +(

∆x − a∆t

∆x

)un

I (C.4)

or

un+1I = un

I − a∆t

∆x(un

I − unI−1) (C.5)

Similarly, the Lax–Wendroff [3] scheme can be derived by computing thevalue un

k by adopting a quadratic interpolation (P = 2) based on a symmetricthree–point stencil around I. With the use of (C.2) and (C.3), once againI0 = I − 1 and ∆I = 1, one gets

un+1I = un

k =(

a∆t

∆x

)(∆x + a∆t

∆x

)un

I−1 +(

∆x − a∆t

∆x

)

(∆x + a∆t

∆x

)un

I +(

∆x − a∆t

∆x

)(−a∆t

∆x

)un

I+1

(C.6)

FD as Characteristic Tracing and Solution Interpolation 354

or after some algebraic manipulation,

un+1I = un

I − a∆t

∆x(un

I+1 − unI−1) +

a2∆t2

∆x2(un

I+1 − 2unI + un

I−1) (C.7)

Many other finite difference representations can be determined using poly-nomial fitting and characteristic tracking, for instance the Lax–Friedrich [2]and the Warming–Beam [5] schemes consider the stencils (I − 1, I + 1) and(I − 2, I − 1, I). This is equivalent to taking (I0 = I − 1, ∆I = 2) and(I0 = I − 2, ∆I = 1) in equation (C.3) respectively.

The effect of a linear and a quadratic interpolation of a typical smoothfunction can be seen in figure C.2(a), where the increase in the order of theinterpolated polynomial leads to an improved representation of the function.However, this is not necessarily true, since a higher–order interpolant, withadditional constrained points, tends to oscillate between the known valueswhen used to fit a function with strong gradients, see figure C.2(b).

(a) (b)

Exact

Low-order

High-order

Exact

Low-order

High-order

Figure C.2: Polynomial interpolation effects. (a) Smooth function; (b) functionwith sharp gradient.

It is also attested by anyone who has ever experimented that to use ex-trapolation is extremely hazardous. Interestingly enough, is the observationthat the CFL condition (5.45) assures no extrapolation during the polyno-mial fitting, which otherwise would make the resultant finite difference schemeunstable.

Bibliography

[1] C. HIRSCH. Numerical Computation of Internal and External Flows, vol-ume 2. John Wiley & Sons, 1990.

[2] P.D. LAX. Weak Solutions of Nonlinear Hyperbolic Equations and TheirNumerical Computation. Comm. Pure and Applied Mathematics, 7:159–193, 1954.



[5] R.F. WARMING and R.M. BEAM. Upwind Second Order DifferenceSchemes and Applications in Aerodynamics Flows. AIAA J., 14:1241–1249, 1976.

[6] O.C. ZIENKIEWICZ and K. MORGAN. Finite Elements and Approxima-tion. John Wiley & Sons, 1983.

355

Appendix D

Determination of theEdge–Based Discrete Equation

In this appendix we deduce the form (7.12) presented in chapter 7 for theGalerkin finite element formulation when an edge–based data structure isadopted. The expressions for the computation of the edge weighting coeffi-cients Cj

IISand boundary face coefficients Df , given in equation (7.13), and

the properties of these coefficients, equation (7.14), are also demonstrated.

Consider the patch of elements shown in figure D.1, where the sides IIS

and I3I lie on the boundary. The right hand side of the discrete statementgiven in (7.6), after introducing (7.9) and (7.10), can be rewritten as

RHSI =mI∑

S=1

∑

E∈IIS

[ΩE

3

∂NI

∂xj

]

E

(F Ij+ F IS

j)

−∑

E∈I

[ΩE

3

∂NI

∂xj

]

E

(F Ij) −

∑

B∈I

[ΓB

6(2F

n

I + Fn

J)]

(D.1)

with the notation already explained in chapter 7.

We consider only the direction x1 and drop the subscript and superscript1 during the following demonstration. Using Green’s theorem and the factthat the shape function NI is zero on the sides ISI1, I1I2 and I2I3, due to thecompact nature in finite element formulation, we have that

∫

Ω

∂NI

∂xdΩ =

∫

ΓNIn dΓ or

∑

E∈I

[ΩE

∂NI

∂x

]

E

=ΓIIS

2nIIS

+ΓI3I

2nI3I (D.2)

Edge–Based Discrete Equation 357

I

Ω2

Ω3

Ω1

I1

I3

I2

Ι S

Figure D.1: Typical patch of triangular elements sharing a boundary node I.

In order to further simplify the development, consider only the contributionsof the boundary face IIS in equation (D.1). Introducing (D.2) into (D.1) wecan write

RHS(IIS)I =

[ΩE1

3

∂NI

∂x

∣∣∣∣E1

](F I + F IS

) −[nIIS

ΓIIS

6

](F I)

−[nIIS

ΓIIS

6

](2F I + F IS

)

(D.3)

After adding and subtracting to equation (D.3) the term

[nIIS

ΓIIS

12

](F I + F IS

) (D.4)

we can easily rearrange (D.3) as

RHS(IIS)I = −1

2

[nIIS

ΓIIS

6− 2ΩE1

3

∂NI

∂x

∣∣∣∣E1

](F I + F IS

)

+[−ΓIIS

12

]nIIS

(4F I + 2F IS+ F I − F IS

)

(D.5)

where the terms inside the square brackets represent the weighting coefficientsC and D for edge IIS in direction x. Exactly the same arguments can beused for direction y (x2), which enables us to write the general forms given inequations (7.12) and (7.13).

Edge–Based Discrete Equation 358

Referring once more to figure D.1, but now considering a generic directionj, the property of the weights given in equation (7.14)(a) can be verified if werewrite it as

RHS(IIS)I = −2

3

(2∑

E∈I

ΩE

[∂NI

∂xj

]

E

)+

1

3

Γj

IIS

2nj

IIS+

ΓjI3I

2nj

I3I

−

−ΓjIIS

2nj

IIS− Γj

I3I

2nj

I3I

= 0

(D.6)

in which the coefficient definition given in (7.13) were used. This is equivalentto

−4

3

∫

Ω

∂NI

∂xdΩ +

4

3

∫

ΓNIn dΓ = 0 (D.7)

which is true from Green’s theorem, (D.2).

The asymmetry property of the edge weights represented in equation(7.14)(b) can be demonstrated using geometrical arguments. Observe thatthe derivatives ∂NI/∂xj |E and the outward normal vector nIIS

have the sameabsolute value as ∂NIS

/∂xj |E and nISI respectively, but opposite sign. There-fore, the coefficients Cj

IISand Cj

ISI , defined according to (7.13) have the sameabsolute value and opposite sign.

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