Riemann Solvers and Numerical Methodsfor Fluid Dynamics

Third Edition

Eleuterio F. Toro

Riemann Solversand Numerical Methodsfor Fluid Dynamics

A Practical Introduction

Third Edition

123

ISBN 978-3-540-25202-3 e-ISBN 978-3-540-49834-6DOI 10.1007/978-3-540-49834-6Springer Dordrecht Heidelberg London New York

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Professor Eleuterio F. ToroUniversity of TrentoItalyeleuteriofrancisco.toro@unitn.it

In memory of two remarkable persons:

my mother Olga Munoz de Toro (Chile, 1930–2003) and

my father–in–law Fritz Freund (Germany, 1926–2008).

Preface to the First Edition

In 1917, the British scientist L. F. Richardson made the first reportedattempt to predict the weather by solving partial differential equations nu-merically, by hand! It is generally accepted that Richardson’s work, thoughunsuccessful, marked the beginning of Computational Fluid Dynamics (CFD),a large branch of scientific computing today. His work had the four distinguish-ing characteristics of CFD: a practical problem to solve, a mathematical

model to represent the problem in the form of a set of partial differential equa-tions, a numerical method and a computer, human beings in Richardson’scase. Eighty years on and these four elements remain the pillars of modernCFD. It is therefore not surprising that the generally accepted definition ofCFD as the science of computing numerical solutions to partial differential orintegral equations that are models for fluid flow phenomena, closely embodiesRichardson’s work.

Computers have, since Richardson’s era, developed to unprecedented lev-els and at an ever decreasing cost. The range of application areas giving riseto practical problems to be solved numerically has increased dramati-cally. In addition to the traditional demands from meteorology, oceanogra-phy, some branches of physics and from a range of engineering disciplines,there are at present fresh demands from a dynamic and fast–moving man-ufacturing industry, whose traditional build–test–fix approach is rapidly be-ing replaced by the use of quantitative methods, at all levels. The need fornew materials and for decision–making under environmental constraints areincreasing sources of demands for mathematical modelling, numerical algo-rithms and high–performance computing. mathematical models have im-proved, though the basic equations of continuum mechanics, already availablemore than a century before Richardson’s first attempts at CFD, are still thebases for modelling fluid flow processes. Progress is required at the level ofthermodynamics, equations of state, and advances into the modelling of non–equilibrium and multiphase flow phenomena. numerical methods are per-haps the success story of the last eighty years, the last twenty being perhapsthe most productive. This success is firmly based on the pioneering works of

VIII Preface to the First Edition

scientists such as von Neumann, whose research on stability explained andresolved the difficulties experienced by Richardson. This success would havebeen impossible without the contributions from Courant, Friedrichs, Richt-myer, Lax, Oleinik, Wendroff, Godunov, Rusanov, van Leer, Harten, Roe,Osher, Colella, Yee, and many others. The net result is: more accurate, moreefficient, more robust and more sophisticated numerical methods are availablefor ambitious practical applications today.

Due to the massive demands on CFD and the level of sophistication ofnumerical methods, new demands on education and training of the scientistsand engineers of the present and the future have arisen. This book is anattempt to contribute to the training and education in numerical methods forfluid dynamics and related disciplines.

The contents of this book were developed over a period of many years ofinvolvement in research on numerical methods, application of the methodsto solve practical problems and teaching scientist and engineers at the post–graduate level. The starting point was a module for a Masters Course in Com-putational Fluid Dynamics at the College of Aeronautics, Cranfield, UK. Thematerial was also part of short courses and lectures given at Cranfield, UK; theErnst Mach Institute, Freiburg, Germany; the Shock Wave Research Centre,Tohoku University, Sendai, Japan; the Department of Mathematics and theDepartment of Civil and Environmental Engineering, University of Trento,Italy; the Department of Mathematics, Technical University Federico SantaMaria, Chile; the Department of Mechanics, Technical University of Aachen,Germany; and the Manchester Metropolitan University (MMU), Manchester,UK.

This book is about modern shock–capturing numerical methods for solv-ing time–dependent hyperbolic conservation laws, with smooth and discon-tinuous solutions, in general multidimensional geometries. The approach iscomprehensive, practical and, in the main, informal. All necessary items ofinformation for the practical implementation of all methods studied here, areprovided in detail. All methods studied are illustrated through practical nu-merical examples; numerical results are compared with exact solutions and insome cases with reliable experimental data.

Most of the book is devoted to a coherent presentation of Godunov meth-ods. The developments of Godunov’s approach over the last twenty years haveled to a mature numerical technology, that can be utilised with confidence tosolve practical problems in established as well as new areas of application.Godunov methods rely on the solution of the Riemann problem. The exactsolution is presented in detail, so as to aid the reader in applying the solutionmethodology to other hyperbolic systems. We also present a variety of approx-imate Riemann solvers; again, the amount of detail supplied will hopefully aidthe reader who might want to apply the methodologies to solve other prob-lems. Other related methods such as the Random Choice Method and the FluxVector Splitting Method are also included. In addition, we study centred (non–upwind) shock–capturing methods. These schemes are much less sophisticated

Preface to the First Edition IX

than Godunov methods, and offer a cheap and simple alternative. High–orderextensions of these methods are constructed for scalar PDEs, along with theirTotal Variation Diminishing (TVD) versions. Most of these TVD methodsare then extended to one–dimensional non–linear systems. Techniques to dealwith PDEs with source terms are also studied, as are techniques for multidi-mensional systems in general geometries.

The presentation of the schemes for non–linear systems is carried outthrough the time–dependent Euler equations of gas dynamics. Having readthe relevant chapters/sections, the reader will be sufficiently well equipped toextend the techniques to other hyperbolic systems, and to advection–reaction–diffusion PDEs.

There are at least two ways of utilising this book. First, it can be used asa means for self–study. In the presentation of the concepts, the emphasis hasbeen placed on clarity, sometimes sacrificing mathematical rigour. The typicalreader in mind is a graduate student in a department of engineering, physics,applied mathematics or computer science, embarking on a research topic thatinvolves the implementation of numerical methods, from first principles, tosolve advection–reaction–diffusion problems. The contents of this book mayalso be useful to numerical analysts beginning their research on algorithms,as elementary background reading. Such users may benefit from a compre-hensive self–study of all the contents of the book, in a period of about twomonths, perhaps including the practical implementation and testing of mostnumerical methods presented. Another class of readers who may benefit fromself–studying this book are scientists and engineers in industry and researchlaboratories. At the cost of some repetitiveness, each chapter is almost self–contained and has plenty of cross–referencing, so that the reader may decideto start reading this book in the middle or jump to the last chapter.

This book can also be used as a teaching aid. Academics involved in theteaching of numerical methods may find this work a useful reference book.Selected chapters or sections may well form the bases for a final year under-graduate course on numerical methods for PDEs. In a mathematics or com-puter science department, the contents may include: some sections of chapter1, chapters 2, 5, 13, some sections of chapter 14, chapter 15 and some sectionsof chapter 16. In a department of engineering or physics, one may includechapters 3, 4, 6, 7, 8, 9, 10, 11, 12, 15, 16 and 17. A postgraduate course mayinvolve most of the contents of this book, assuming perhaps a working knowl-edge of compressible fluid dynamics. Short courses for training engineers andscientists in industry and research laboratories can also be based on most ofthe contents of this book.

Eleuterio ToroManchester, UKMarch 1997.

X Preface to the First Edition

ACKNOWLEDGEMENTS FOR FIRST EDITION

The author is indebted to many colleagues who over the years have kindlyarranged short and extended visits to their establishments, have organisedseminars, workshops and short courses. These events have strongly influ-enced the development of this book. Special thanks are due to Dr. W. Heilig(Freiburg, Germany); Professors V. Casulli and A. Armanini (Trento, Italy);Professor K. Takayama (Sendai, Japan); Professor J. Figueroa (Valparaiso,Chile) and Professor J. Ballmann (Aachen, Germany). The final stages of thetyping of the material were carried out while the author was at MMU, Manch-ester, UK; the support provided is gratefully acknowledged. Thanks are alsodue to my former and current PhD students, for their comments on my lec-tures, on the contents of this book and for their contribution to research inrelevant areas. Special thanks are due to Stephen Billett (Centre for Atmo-spheric Science, University of Cambridge, UK), Caroline Lowe (Department ofAerospace Science, Cranfield University, UK), Nikolaos Nikiforakis (DAMTP,University of Cambridge, UK), Ed Boden (Cranfield and MMU), Ms WeiHu, Mathew Ivings and Richard Millington (MMU). Thanks are due to myson Marcelo, who helped with the type–setting, and to David Ingram andJohn Nuttall for their help with the intricacies of Latex. The author thanksSpringer–Verlag, and specially Miss Erdmuthe Raufelder in Heidelberg, fortheir professional assistance.

The author gratefully acknowledges useful feedback on the manuscriptfrom Stephen Billett (University of Cambridge, UK), John Clarke (Cran-field University, UK), Jean–Marc Moschetta (SUPAERO, Toulouse, France),Claus–Dieter Munz (University of Stuttgart, Germany), Jack Pike (Bedford,UK), Ning Qin (Cranfield University, UK), Tsutomu Saito (Cray Japan),Charles Saurel (University of Provence, France), Trevor Randall (MMU,Manchester, UK), Peter Sweby (University of Reading, UK), Marcelo Toro(IBM Glasgow, UK), Helen Yee (NASA Ames, USA) and Clive Woodley(DERA, Fort Halstead, UK).

The permanent encouragement from Brigitte has been an immensely valu-able support during the writing of this book. Thank you Brigitte!

Preface to the Third Edition

More than a decade has elapsed since the publication of the first edition of thisbook. During this period the response from readers the world over has beenoverwhelming, from students and academics to senior researchers. This decadehas also witnessed a significant increase in the use of numerical methods, notonly in the traditional areas such as physics and industrial processes, but alsoin biology, economics, social sciences and in inter–disciplinary research areas.We also observe a new trend in mathematical modelling and numerical simu-lation. Numerical methods are steadily moving from being a simulation toolfor engineering design in technology, to being an indispensable instrument inscience, for studying and understanding phenomena of the most varied kind.The expectation is that a simulation will represent the solution of the actualmathematical model. Such expectation implies the need for more and morehigh–quality research on new and more accurate numerical methods and theneed for better training of scientists at the undergraduate and post–graduatelevels at universities and higher education institutions. I expect that this newedition of the book will continue to play a role in such endeavours.

In this edition I have included three new chapters, chapters 18 to 20. Chapter18 is about a multi–dimensional extension of the centred FORCE flux stud-ied in Chapter 7, for one–dimensional systems; in a sense, this new chapteris a response to the increasing role of the so–called centred methods. Thesehave the advantage of avoiding the direct solution of the classical Riemannproblem, in the conventional manner. As a result, the applicability of thesecentred schemes is more general than that of conventional upwind methods;this feature is specially useful when having to solve complicated systems, forwhich the solution of the Riemann problem may be difficult or impossibleto obtain. The new FORCE scheme applies to two and three space dimen-sions on general structured and unstructured meshes and can be extended tohigh order of accuracy in space and time in the frameworks of finite volumeand discontinuous Galerkin finite element methods. The second new addition,chapter 19, is about the high–order, or generalized, Riemann problem, the

XII Preface to the Third Edition

Cauchy problem for hyperbolic balance laws whose initial conditions are twosmooth vectors separated by a discontinuity at the origin. The solution of thegeneralized Riemann problem serves as the building block for constructingfully discrete, one–step Godunov–type schemes of arbitrary order of accuracyin both space and time. This chapter is effectively a generalization of the ma-terial of this book studied in previous chapters and responds to the generaltrend of improving the accuracy, in space and time, of numerical methods forsolving evolutionary partial differential equations. These high–order numericalschemes can be constructed in the frameworks of finite volume and discon-tinuous Galerkin finite element methods. The third new chapter, chapter 20,contains an introduction to these high–order methods in the framework offinite volumes.

This edition of the book contains a substantially revised version of the HLLCRiemann solver of chapter 10. This responds to many communications re-ceived from readers and to new developments of the technique and its use forvery ambitious scientific and technological applications. Some modificationsto chapter 21 have also been carried out, as well general corrections to errorspointed out by readers.

This book, in spite of being introductory in nature, continues to be a bookused mainly by researchers, and is perhaps too advanced for teaching under-graduate students. As a result, a new more elementary book is being writtenin collaboration with Enrico Bertolazzi and Gianluca Vignoli. This new book,to be published by Springer in 2009, is specifically designed for teaching un-dergraduate students in Science and Engineering, with plenty of exercises,case studies and miniprojects. On the other hand, in order to respond to re-search needs for better and more sophisticated numerical methods, we arecurrently preparing a new book, in collaboration with Claus–Dieter Munz,Vladimir Titarev and Michael Dumbser. This book will deal with advanced,high–order, finite volume and discontinuous Galerkin numerical methods forstructured and unstructured meshes in multiple space dimensions, to be pub-lished by Springer in 2009.

I gratefully acknowledge the contribution of some collaborators to the prepara-tion of this third edition. In particular I thank two of my former PhD students,Dr. Vladimir Titarev, now at Cranfield University, UK and Dr. Cristobal Cas-tro, now at the Technical University of Munich, Germany. I also thank twoformer post–doctoral fellows, Dr. Martin Kaser, now at the Technical Uni-versity of Munich, Germany and Dr. Michael Dumbser, now a colleague inmy group at Trento University, Italy. Thanks are also due to colleague EnricoBertolazzi and to visiting scholars Marıa Nofuentes (Universidad de Cordoba,Spain) and Arturo Hidalgo (Universidad Politecnica de Madrid, Spain), whokindly helped in various ways in the preparation of the material.

Preface to the Third Edition XIII

I find it also appropriate to recall that the initial stages in the preparationof the first edition of this book took place at Cranfield University, formerlyCranfield Institute of Techology, United Kingdom. During the late 80s andearly 90s, a very dynamic and stimulating group of scientists from variousbackgrounds, existed in the Aerodynamics Department, within the School ofAerospace Sciences. Those years at Cranfield greatly influenced the contentsand scope of this book, for which I thank former senior and junior colleaguesand students, John Clarke (FRS), Phil Roe, Jack Pike, Smadar Karni, JamesQuirk, Nikos Nikiforakis, Stephen Billett, Alan Dowes, Caroline Lowe, WilliamSpeares, Michael Spruce, Ed Boden, Mauricio Caceres, and many others.

Finally, I thank Springer Verlag for their permanent and positive encourage-ment and effective support.

Eleuterio ToroUniversity of Trento, ItalyDecember 2008

Contents

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI

1 The Equations of Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Conservation–Law Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Other Compact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Thermodynamic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Units of Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Equations of State (EOS) . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Other Variables and Relations . . . . . . . . . . . . . . . . . . . . . . 71.2.4 Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.5 Covolume and van der Waal Gases . . . . . . . . . . . . . . . . . . 13

1.3 Viscous Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Integral Form of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.1 Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5.2 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5.3 Conservation of Momentum. . . . . . . . . . . . . . . . . . . . . . . . . 211.5.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6 Submodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6.1 Summary of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . 251.6.2 Flow with Area Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6.3 Axi–Symmetric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.6.4 Cylindrical and Spherical Symmetry . . . . . . . . . . . . . . . . . 291.6.5 Plain One–Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . 291.6.6 Steady Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 311.6.7 Viscous Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . 331.6.8 Free–Surface Gravity Flow. . . . . . . . . . . . . . . . . . . . . . . . . . 331.6.9 The Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . . . 35

XVI Contents

1.6.10 Incompressible Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . 381.6.11 The Artificial Compressibility Equations . . . . . . . . . . . . . 39

2 Notions on Hyperbolic Partial Differential Equations . . . . . . 412.1 Quasi–Linear Equations: Basic Concepts . . . . . . . . . . . . . . . . . . . 422.2 The Linear Advection Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Characteristics and the General Solution . . . . . . . . . . . . . 472.2.2 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3 Linear Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3.1 Diagonalisation and Characteristic Variables . . . . . . . . . . 512.3.2 The General Initial–Value Problem . . . . . . . . . . . . . . . . . . 522.3.3 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3.4 The Riemann Problem for Linearised Gas Dynamics . . . 582.3.5 Some Useful Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4.1 Integral Forms of Conservation Laws . . . . . . . . . . . . . . . . . 622.4.2 Non–Linearities and Shock Formation . . . . . . . . . . . . . . . . 662.4.3 Characteristic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.4.4 Elementary–Wave Solutions of the Riemann Problem . . 83

3 Some Properties of the Euler Equations . . . . . . . . . . . . . . . . . . . 873.1 The One–Dimensional Euler Equations . . . . . . . . . . . . . . . . . . . . . 87

3.1.1 Conservative Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 873.1.2 Non–Conservative Formulations . . . . . . . . . . . . . . . . . . . . . 913.1.3 Elementary Wave Solutions of the Riemann Problem . . . 94

3.2 Multi–Dimensional Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.1 Two–Dimensional Equations in Conservative Form . . . . 1043.2.2 Three–Dimensional Equations in Conservative Form . . . 1083.2.3 Three–Dimensional Primitive Variable Formulation . . . . 1093.2.4 The Split Three–Dimensional Riemann Problem . . . . . . . 111

3.3 Conservative Versus Non–Conservative Formulations . . . . . . . . . 112

4 The Riemann Problem for the Euler Equations . . . . . . . . . . . . 1154.1 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2 Equations for Pressure and Particle Velocity . . . . . . . . . . . . . . . . 119

4.2.1 Function fL for a Left Shock . . . . . . . . . . . . . . . . . . . . . . . . 1204.2.2 Function fL for Left Rarefaction . . . . . . . . . . . . . . . . . . . . 1224.2.3 Function fR for a Right Shock . . . . . . . . . . . . . . . . . . . . . . 1234.2.4 Function fR for a Right Rarefaction . . . . . . . . . . . . . . . . . 124

4.3 Numerical Solution for Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.1 Behaviour of the Pressure Function . . . . . . . . . . . . . . . . . . 1254.3.2 Iterative Scheme for Finding the Pressure . . . . . . . . . . . . 1274.3.3 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.4 The Complete Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.5 Sampling the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Contents XVII

4.5.1 Left Side of Contact: S = x/t ≤ u∗ . . . . . . . . . . . . . . . . . . 1374.5.2 Right Side of Contact: S = x/t ≥ u∗ . . . . . . . . . . . . . . . . . 137

4.6 The Riemann Problem in the Presence of Vacuum . . . . . . . . . . . 1394.6.1 Case 1: Vacuum Right State . . . . . . . . . . . . . . . . . . . . . . . . 1404.6.2 Case 2: Vacuum Left State . . . . . . . . . . . . . . . . . . . . . . . . . 1424.6.3 Case 3: Generation of Vacuum . . . . . . . . . . . . . . . . . . . . . . 142

4.7 The Riemann Problem for Covolume Gases . . . . . . . . . . . . . . . . . 1434.7.1 Solution for Pressure and Particle Velocity. . . . . . . . . . . . 1444.7.2 Numerical Solution for Pressure . . . . . . . . . . . . . . . . . . . . . 1474.7.3 The Complete Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1474.7.4 Solution Inside Rarefactions . . . . . . . . . . . . . . . . . . . . . . . . 148

4.8 The Split Multi–Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . 1494.9 FORTRAN Program for Exact Riemann Solver . . . . . . . . . . . . . 151

5 Notions on Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.1 Discretisation: Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . 163

5.1.1 Approximation to Derivatives . . . . . . . . . . . . . . . . . . . . . . . 1645.1.2 Finite Difference Approximation to a PDE . . . . . . . . . . . 165

5.2 Selected Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.2.1 The First Order Upwind Scheme . . . . . . . . . . . . . . . . . . . . 1685.2.2 Other Well–Known Schemes . . . . . . . . . . . . . . . . . . . . . . . . 172

5.3 Conservative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.3.2 Godunov’s First–Order Upwind Method . . . . . . . . . . . . . . 1775.3.3 Godunov’s Method for Burgers’s Equation . . . . . . . . . . . . 1815.3.4 Conservative Form of Difference Schemes . . . . . . . . . . . . . 184

5.4 Upwind Schemes for Linear Systems . . . . . . . . . . . . . . . . . . . . . . . 1875.4.1 The CIR Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.4.2 Godunov’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.5 Sample Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.5.1 Linear Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.5.2 The Inviscid Burgers Equation . . . . . . . . . . . . . . . . . . . . . . 196

5.6 FORTRAN Program for Godunov’s Method . . . . . . . . . . . . . . . . 196

6 The Method of Godunov for Non–linear Systems . . . . . . . . . . 2136.1 Bases of Godunov’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2136.2 The Godunov Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166.3 Godunov’s Method for the Euler Equations . . . . . . . . . . . . . . . . . 218

6.3.1 Evaluation of the Intercell Fluxes . . . . . . . . . . . . . . . . . . . . 2196.3.2 Time Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 2256.4.1 Numerical Results for Godunov’s Method . . . . . . . . . . . . 2266.4.2 Numerical Results from Other Methods . . . . . . . . . . . . . . 228

XVIII Contents

7 Random Choice and Related Methods . . . . . . . . . . . . . . . . . . . . . 2377.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.2 RCM on a Non–Staggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

7.2.1 The Scheme for Non–Linear Systems . . . . . . . . . . . . . . . . . 2397.2.2 Boundary Conditions and the Time Step Size . . . . . . . . . 243

7.3 A Random Choice Scheme of the Lax–Friedrichs Type . . . . . . . 2447.3.1 Review of the Lax–Friedrichs Scheme . . . . . . . . . . . . . . . . 2447.3.2 The Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.4 The RCM on a Staggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.4.1 The Scheme for Non–Linear Systems . . . . . . . . . . . . . . . . . 2477.4.2 A Deterministic First–Order Centred Scheme (force) . 2477.4.3 Analysis of the force Scheme . . . . . . . . . . . . . . . . . . . . . . 249

7.5 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2507.5.1 Van der Corput Pseudo–Random Numbers . . . . . . . . . . . 2507.5.2 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.5.3 Propagation of a Single Shock . . . . . . . . . . . . . . . . . . . . . . . 253

7.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

8 Flux Vector Splitting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2658.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2658.2 The Flux Vector Splitting Approach . . . . . . . . . . . . . . . . . . . . . . . 266

8.2.1 Upwind Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2668.2.2 The FVS Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.3 FVS for the Isothermal Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2708.3.1 Split Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.3.2 FVS Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

8.4 FVS Applied to the Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 2738.4.1 Recalling the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2748.4.2 The Steger–Warming Splitting . . . . . . . . . . . . . . . . . . . . . . 2768.4.3 The van Leer Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.4.4 The Liou–Steffen Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

8.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.5.1 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.5.2 Results for Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.5.3 Results for Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.5.4 Results for Test 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.5.5 Results for Test 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2828.5.6 Results for Test 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

9 Approximate–State Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . 2939.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.2 The Riemann Problem and the Godunov Flux . . . . . . . . . . . . . . 294

9.2.1 Tangential Velocity Components . . . . . . . . . . . . . . . . . . . . 2969.2.2 Sonic Rarefactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Contents XIX

9.3 Primitive Variable Riemann Solvers (PVRS) . . . . . . . . . . . . . . . . 2979.4 Approximations Based on the Exact Solver . . . . . . . . . . . . . . . . . 301

9.4.1 A Two–Rarefaction Riemann Solver (TRRS) . . . . . . . . . . 3019.4.2 A Two–Shock Riemann Solver (TSRS) . . . . . . . . . . . . . . . 303

9.5 Adaptive Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3049.5.1 An Adaptive Iterative Riemann Solver (AIRS) . . . . . . . . 3049.5.2 An Adaptive Noniterative Riemann Solver (ANRS) . . . . 305

9.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

10 The HLL and HLLC Riemann Solvers . . . . . . . . . . . . . . . . . . . . . 31510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.2 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

10.2.1 The Godunov Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31710.2.2 Integral Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

10.3 The HLL Approximate Riemann Solver . . . . . . . . . . . . . . . . . . . . 32010.4 The HLLC Approximate Riemann Solver . . . . . . . . . . . . . . . . . . . 322

10.4.1 Useful Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210.4.2 The HLLC Flux for the Euler Equations . . . . . . . . . . . . . 32410.4.3 Multidimensional and Multicomponent Flow . . . . . . . . . . 326

10.5 Wave–Speed Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32710.5.1 Direct Wave Speed Estimates . . . . . . . . . . . . . . . . . . . . . . . 32810.5.2 Pressure–Based Wave Speed Estimates . . . . . . . . . . . . . . . 329

10.6 Summary of HLLC Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33110.7 Contact Waves and Passive Scalars . . . . . . . . . . . . . . . . . . . . . . . . 33310.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.9 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

11 The Riemann Solver of Roe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34511.1 Bases of the Roe Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

11.1.1 The Exact Riemann Problem and the Godunov Flux . . . 34611.1.2 Approximate Conservation Laws . . . . . . . . . . . . . . . . . . . . 34711.1.3 The Approximate Riemann Problem and the Intercell

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34911.2 The Original Roe Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

11.2.1 The Isothermal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 35211.2.2 The Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

11.3 The Roe–Pike Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35811.3.1 The Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35811.3.2 The Isothermal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 35911.3.3 The Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

11.4 An Entropy Fix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36611.4.1 The Entropy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36611.4.2 The Harten–Hyman Entropy Fix . . . . . . . . . . . . . . . . . . . . 36711.4.3 The Speeds u∗, a∗L, a∗R . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

11.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 372

XX Contents

11.5.1 The Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37211.5.2 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

11.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

12 The Riemann Solver of Osher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.1 Osher’s Scheme for a General System . . . . . . . . . . . . . . . . . . . . . . 378

12.1.1 Mathematical Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37812.1.2 Osher’s Numerical Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38012.1.3 Osher’s Flux for the Single–Wave Case . . . . . . . . . . . . . . . 38112.1.4 Osher’s Flux for the Inviscid Burgers Equation . . . . . . . . 38312.1.5 Osher’s Flux for the General Case . . . . . . . . . . . . . . . . . . . 384

12.2 Osher’s Flux for the Isothermal Equations . . . . . . . . . . . . . . . . . . 38512.2.1 Osher’s Flux with P–Ordering . . . . . . . . . . . . . . . . . . . . . . 38612.2.2 Osher’s Flux with O–Ordering . . . . . . . . . . . . . . . . . . . . . . 389

12.3 Osher’s Scheme for the Euler Equations . . . . . . . . . . . . . . . . . . . 39212.3.1 Osher’s Flux with P–Ordering . . . . . . . . . . . . . . . . . . . . . . 39312.3.2 Osher’s Flux with O–Ordering . . . . . . . . . . . . . . . . . . . . . . 39712.3.3 Remarks on Path Orderings . . . . . . . . . . . . . . . . . . . . . . . . 40212.3.4 The Split Three–Dimensional Case . . . . . . . . . . . . . . . . . . 403

12.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 40412.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

13 High–Order and TVD Methods for Scalar Equations . . . . . . 41313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41313.2 Basic Properties of Selected Schemes . . . . . . . . . . . . . . . . . . . . . . . 415

13.2.1 Selected Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41513.2.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41713.2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

13.3 WAF–Type High Order Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 42013.3.1 The Basic waf Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42013.3.2 Generalisations of the waf Scheme . . . . . . . . . . . . . . . . . . 423

13.4 MUSCL–Type High–Order Methods . . . . . . . . . . . . . . . . . . . . . . . 42613.4.1 Data Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42613.4.2 The MUSCL–Hancock Method (MHM) . . . . . . . . . . . . . . 42913.4.3 The Piece–Wise Linear Method (PLM) . . . . . . . . . . . . . . . 43213.4.4 The Generalised Riemann Problem (GRP) Method . . . . 43413.4.5 Slope–Limiter Centred (slic) Schemes . . . . . . . . . . . . . . . 43613.4.6 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43913.4.7 Semi–Discrete Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43913.4.8 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

13.5 Monotone Schemes and Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 44013.5.1 Monotone Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44013.5.2 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44313.5.3 Monotone Schemes and Godunov’s Theorem . . . . . . . . . . 44713.5.4 Spurious Oscillations and High Resolution . . . . . . . . . . . . 448

Contents XXI

13.5.5 Data Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44913.6 Total Variation Diminishing ( TVD) Methods . . . . . . . . . . . . . . . 451

13.6.1 The Total Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45213.6.2 TVD and Monotonicity Preserving Schemes . . . . . . . . . . 453

13.7 Flux Limiter Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45613.7.1 TVD Version of the waf Method . . . . . . . . . . . . . . . . . . . 45613.7.2 The General Flux–Limiter Approach . . . . . . . . . . . . . . . . . 46413.7.3 TVD Upwind Flux Limiter Schemes . . . . . . . . . . . . . . . . . 46913.7.4 TVD Centred Flux Limiter Schemes . . . . . . . . . . . . . . . . 474

13.8 Slope Limiter Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48013.8.1 TVD Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48013.8.2 Construction of TVD Slopes . . . . . . . . . . . . . . . . . . . . . . . . 48113.8.3 Slope Limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48213.8.4 Limited Slopes Obtained from Flux Limiters . . . . . . . . . . 484

13.9 Extensions of TVD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48613.9.1 TVD Schemes in the Presence of Source Terms . . . . . . . 48613.9.2 TVD Schemes in the Presence of Diffusion Terms . . . . . 486

13.10Numerical Results for Linear Advection . . . . . . . . . . . . . . . . . . . . 487

14 High–Order and TVD Schemes for Non–Linear Systems . . . 49314.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49314.2 CFL and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.3 Weighted Average Flux (waf) Schemes . . . . . . . . . . . . . . . . . . . . 496

14.3.1 The Original Version of waf . . . . . . . . . . . . . . . . . . . . . . . . 49614.3.2 A Weighted Average State Version . . . . . . . . . . . . . . . . . . 49814.3.3 Rarefactions in State Riemann Solvers . . . . . . . . . . . . . . . 49914.3.4 TVD Version of waf Schemes . . . . . . . . . . . . . . . . . . . . . . . 50114.3.5 Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50314.3.6 Summary of the waf Method . . . . . . . . . . . . . . . . . . . . . . . 503

14.4 The MUSCL–Hancock Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50414.4.1 The Basic Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50414.4.2 A Variant of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 50614.4.3 TVD Version of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . 50714.4.4 Summary of the MUSCL–Hancock Method . . . . . . . . . . . 510

14.5 Centred TVD Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51114.5.1 Review of the force Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 51214.5.2 A Flux Limiter Centred (flic) Scheme . . . . . . . . . . . . . . . 51214.5.3 A Slope Limiter Centred (slic) Scheme . . . . . . . . . . . . . . 514

14.6 Primitive–Variable Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51514.6.1 Formulation of the Equations and Primitive Schemes . . 51514.6.2 A waf–Type Primitive Variable Scheme . . . . . . . . . . . . . . 51714.6.3 A MUSCL–Hancock Primitive Scheme . . . . . . . . . . . . . . . 52014.6.4 Adaptive Primitive–Conservative Schemes . . . . . . . . . . . . 522

14.7 Some Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52314.7.1 Upwind TVD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

XXII Contents

14.7.2 Centred TVD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

15 Splitting Schemes for PDEs with Source Terms . . . . . . . . . . . . 53115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53115.2 Splitting for a Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53315.3 Numerical Methods Based on Splitting . . . . . . . . . . . . . . . . . . . . . 535

15.3.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53515.3.2 Schemes for Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

15.4 Remarks on ODE Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53715.4.1 First–Order Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . 53715.4.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53915.4.3 Implementation Details for Split Schemes . . . . . . . . . . . . . 540

15.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

16 Methods for Multi–Dimensional PDEs . . . . . . . . . . . . . . . . . . . . . 54316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54316.2 Dimensional Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

16.2.1 Splitting for a Model Problem . . . . . . . . . . . . . . . . . . . . . . 54416.2.2 Splitting Schemes for Two–Dimensional Systems . . . . . . 54516.2.3 Splitting Schemes for Three–Dimensional Systems . . . . . 547

16.3 Practical Implementation of Splitting Schemes in ThreeDimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54916.3.1 Handling the Sweeps by a Single Subroutine . . . . . . . . . . 54916.3.2 Choice of Time Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 55116.3.3 The Intercell Flux and the tvd Condition . . . . . . . . . . . . 552

16.4 Unsplit Finite Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 55516.4.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55516.4.2 Accuracy and Stability of Multidimensional Schemes . . . 558

16.5 A Muscl–Hancock Finite Volume Scheme . . . . . . . . . . . . . . . . . . 56116.6 WAF–Type Finite Volume Schemes . . . . . . . . . . . . . . . . . . . . . . . . 563

16.6.1 Two–Dimensional Linear Advection . . . . . . . . . . . . . . . . . . 56416.6.2 Three–Dimensional Linear Advection . . . . . . . . . . . . . . . . 56716.6.3 Schemes for Two–Dimensional Nonlinear Systems . . . . . 57016.6.4 Schemes for Three–Dimensional Nonlinear Systems . . . . 573

16.7 Non–Cartesian Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57416.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57416.7.2 General Domains and Coordinate Transformation . . . . . 57516.7.3 The Finite Volume Method for Non–Cartesian Domains 578

17 Multidimensional Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 58517.1 Explosions and Implosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

17.1.1 Explosion Test in Two–Space Dimensions . . . . . . . . . . . . 58717.1.2 Explosion Test in Three Space Dimensions . . . . . . . . . . . . 590

17.2 Shock Wave Reflection from a Wedge . . . . . . . . . . . . . . . . . . . . . . 591

Contents XXIII

18 FORCE Fluxes in Multiple Space Dimensions . . . . . . . . . . . . . 59718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59718.2 Review of FORCE in One Space Dimension . . . . . . . . . . . . . . . . . 600

18.2.1 FORCE and Related Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 60018.2.2 Monotonicity and Numerical Viscosity . . . . . . . . . . . . . . . 602

18.3 FORCE Schemes on Cartesian Meshes . . . . . . . . . . . . . . . . . . . . . 60518.3.1 The Two–Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 60518.3.2 The Three–Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . 609

18.4 Properties of the FORCE Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 61018.4.1 One–Dimensional Interpretation . . . . . . . . . . . . . . . . . . . . . 61018.4.2 Some Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . 61118.4.3 Analysis in Multiple Space Dimensions . . . . . . . . . . . . . . . 613

18.5 FORCE Schemes on General Meshes . . . . . . . . . . . . . . . . . . . . . . . 61718.6 Sample Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62118.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

19 The Generalized Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . 62519.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62519.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62919.3 The Cauchy–Kowalewski Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 631

19.3.1 Series Expansions and Analytic Functions . . . . . . . . . . . . 63219.3.2 Illustration of the Cauchy–Kowalewski Theorem . . . . . . . 63319.3.3 The Cauchy–Kowalewski Method . . . . . . . . . . . . . . . . . . . . 633

19.4 A Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63519.4.1 The Leading Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63619.4.2 Higher–Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63719.4.3 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64019.4.4 Summary: Numerical Flux and Numerical Source . . . . . . 64019.4.5 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

19.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64219.5.1 The Linear Advection Equation . . . . . . . . . . . . . . . . . . . . . 64319.5.2 Linear Advection with a Source Term . . . . . . . . . . . . . . . . 64519.5.3 Non–Linear Equation with a Source Term . . . . . . . . . . . . 64619.5.4 The Burgers Equation with a Source Term . . . . . . . . . . . 648

19.6 Other Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65119.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

20 The ADER Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65520.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65520.2 The Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

20.2.1 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65720.2.2 The Numerical Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65820.2.3 The Numerical Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65920.2.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

20.3 Second–Order Scheme for a Model Equation . . . . . . . . . . . . . . . . 663

XXIV Contents

20.3.1 Numerical Flux and Numerical Source . . . . . . . . . . . . . . . 66320.3.2 The Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

20.4 Schemes of Arbitrary Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66720.4.1 The Numerical Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66720.4.2 The Numerical Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66820.4.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

20.5 Sample Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66920.5.1 Long–Time Advection of Smooth Profiles . . . . . . . . . . . . . 66920.5.2 Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

20.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

21 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67921.1 Summary of Numerical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 67921.2 Potential Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68121.3 Current Research Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68521.4 The NUMERICA Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

1

The Equations of Fluid Dynamics

In this chapter we present the governing equations for the dynamics ofa compressible material, such as a gas, along with closure conditions in theform of equations of state. Equations of state are statements about the natureof the material in question and require some notions from Thermodynamics.There is no attempt to provide an exhaustive and rigourous derivation of theequations of continuum mechanics; such a task is beyond the scope of thisbook. Instead, we give a fairly self–contained summary of the equations andthe Thermodynamics in a manner that is immediately useful to the mainpurpose of this book, namely the detailed treatment of Riemann solvers andnumerical methods.

The presentation of the equations is unconventional. We first introduce thedifferential form of the Euler equations along with basic physical quantitiesand thermodynamic relations leading to equations of state. Then the effectsof viscous diffusion and heat transfer are added to the Euler equations. Afterthis, the fundamental integral form of the equations is introduced; conven-tionally, this is the starting point for presenting the governing equations. Thischapter contains virtually all of the necessary background on Fluid Dynamicsthat is required for a fruitful study of the rest of the book. It also containsuseful information for those wishing to embark on complex practical applica-tions. A hierarchy of submodels is also presented. This covers four systems ofhyperbolic conservation laws for which Riemann solvers and upwind methodsare directly applicable, namely (i) the time–dependent Euler equations, (ii)the steady supersonic Euler equations, (iii) the shallow water equations and(iv) the artificial compressibility equations associated with the incompress-ible Navier–Stokes equations. Included in the hierarchy are also some simplermodels such as linear systems and scalar conservation laws.

Some remarks on notation are in order. A Cartesian frame of reference(x, y, z) is chosen and the time variable is denoted by t. Transformation toother coordinate systems is carried out using the chain rule in the usual way,see Sect. 16.7.2 of Chap. 16. Any quantity φ that depends on space and timewill be written as φ(x, y, z, t). In most situations the governing equations will

E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 1c© Springer-Verlag Berlin Heidelberg 2009DOI 10.1007/b7976-1 1,

2 1 The Equations of Fluid Dynamics

be partial differential equations (PDEs). Naturally, these will involve partialderivatives for which we use the notation

φt ≡∂φ

∂t, φx ≡ ∂φ

∂x, φy ≡ ∂φ

∂y, φz ≡ ∂φ

∂z.

We also recall some basic notation involving scalars and vectors. The dotproduct of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is the scalarquantity

A · B = a1b1 + a2b2 + a2b3 .

Given a scalar quantity φ that depends on the spatial variables x, y, z thegradient operator ∇ as applied to φ is the vector

grad φ ≡ ∇φ ≡ (∂φ

∂x,∂φ

∂y,∂φ

∂z) .

The divergence operator applies to vectors and the result is a scalar quantity;for a vector A, the divergence of A is

div A ≡ ∇ · A ≡ ∂a1

∂x+

∂a2

∂y+

∂a3

∂z.

1.1 The Euler Equations

In this section we consider the time–dependent Euler equations. These area system of non–linear hyperbolic conservation laws that govern the dynamicsof a compressible material, such as gases or liquids at high pressures, for whichthe effects of body forces, viscous stresses and heat flux are neglected.

There is some freedom in choosing a set of variables to describe the flowunder consideration. A possible choice is the so called primitive variables orphysical variables, namely, ρ(x, y, z, t) = density or mass density, p(x, y, z, t) =pressure, u(x, y, z, t) = x–component of velocity, v(x, y, z, t) = y–componentof velocity, w(x, y, z, t) = z–component of velocity. The velocity vector isV = (u, v, w). An alternative choice is provided by the so called conservedvariables. These are the mass density ρ, the x–momentum component ρu, they–momentum component ρv, the z–momentum component ρw and the totalenergy per unit mass E. Physically, these conserved quantities result natu-rally from the application of the fundamental laws of conservation of mass,Newton’s Second Law and the law of conservation of energy. Computationally,there are some advantages in expressing the governing equations in terms ofthe conserved variables. This gives rise to a large class of numerical methodscalled conservative methods, which will be studied later in this book. We nextstate the equations in terms of the conserved variables under the assumptionthat the quantities involved are sufficiently smooth to allow for the operationof differentiation to be defined. Later we remove this smoothness constraintto allow for solutions containing discontinuities, such as shock waves.

1.1 The Euler Equations 3

1.1.1 Conservation–Law Form

The five governing conservation laws are

ρt + (ρu)x + (ρv)y + (ρw)z = 0 , (1.1)

(ρu)t + (ρu2 + p)x + (ρuv)y + (ρuw)z = 0 , (1.2)

(ρv)t + (ρuv)x + (ρv2 + p)y + (ρvw)z = 0 , (1.3)

(ρw)t + (ρuw)x + (ρvw)y + (ρw2 + p)z = 0 , (1.4)

Et + [u(E + p)]x + [v(E + p)]y + [w(E + p)]z = 0 . (1.5)

Here E is the total energy per unit volume

E = ρ (12V2 + e) , (1.6)

where12V2 =

12V · V =

12(u2 + v2 + w2)

is the specific kinetic energy and e is the specific internal energy. One generallyrefers to the full system (1.1)–(1.5) as the Euler equations, although strictlyspeaking the Euler equations are just (1.2)–(1.4).

The conservation laws (1.1)–(1.5) can be expressed in a very compactnotation by defining a column vector U of conserved variables and flux vectorsF(U), G(U), H(U) in the x, y and z directions, respectively. The equationsnow read

Ut + F(U)x + G(U)y + H(U)z = 0 , (1.7)

with

U =

⎡⎢⎢⎢⎢⎣

ρρuρvρwE

⎤⎥⎥⎥⎥⎦

, F =

⎡⎢⎢⎢⎢⎣

ρuρu2 + p

ρuvρuw

u(E + p)

⎤⎥⎥⎥⎥⎦

,

G =

⎡⎢⎢⎢⎢⎣

ρvρuv

ρv2 + pρvw

v(E + p)

⎤⎥⎥⎥⎥⎦

, H =

⎡⎢⎢⎢⎢⎣

ρwρuwρvw

ρw2 + pw(E + p)

⎤⎥⎥⎥⎥⎦

.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1.8)

It is important to note that F = F(U), G = G(U), H = H(U); that is,the flux vectors are to be regarded as functions of the conserved variablevector U. Any set of PDEs written in the form (1.7) is called a system ofconservation laws. As partial derivatives are involved we say that (1.7) is asystem of conservation laws in differential form. The differential formulationassumes smooth solutions, that is, partial derivatives are assumed to exist.There are other ways of expressing conservation laws in which the smoothnessassumption is relaxed to include discontinuous solutions.

4 1 The Equations of Fluid Dynamics

1.1.2 Other Compact Forms

An even more compact way of expressing equations (1.1)–(1.5) involvestensors. First note that the spatial derivatives in (1.1) can be expressed interms of the divergence operator e.g.

div(ρV) = ∇ · (ρV) = (ρu)x + (ρv)y + (ρw)z .

Thus equation (1.1) for conservation of mass can be written as

ρt + ∇ · (ρV) = 0 . (1.9)

As the divergence operator may also be applied to tensors, the three momen-tum equations for conservation of momentum can be written in compact formas

(ρV)t + ∇ · (ρV ⊗ V + pI) = 0 , (1.10)

where V ⊗ V is the tensor product and I is the unit tensor. These are givenrespectively by

V ⊗ V =

⎛⎝

u2 uv uwvu v2 vwwu wv w2

⎞⎠ , I =

⎛⎝

1 0 00 1 00 0 1

⎞⎠ .

The conservation of energy equation can be written as

Et + ∇ · [(E + p)V] = 0 . (1.11)

In fact the complete system of equations (1.9)–(1.11) can be written in diver-gence form as

Ut + ∇ · H = 0 , (1.12)

where H is the tensor

H =

⎡⎣

ρu ρu2 + p ρvu ρwu u(E + p)ρv ρuv ρv2 + p ρwv v(E + p)ρw ρuw ρvw ρw2 + p w(E + p)

⎤⎦ . (1.13)

Note that the rows of the tensor H are the flux vectors F, G and H, under-stood as row vectors. For computational purposes it is the compact conserva-tive form (1.7)–(1.8) of equations (1.1)–(1.5) that is most useful. In Chap. 3 westudy some mathematical properties of the Euler equations and in Chap. 4 wesolve exactly the Riemann problem for the one–dimensional Euler equationsfor ideal and covolume gases. Numerical methods for the Euler equations arediscussed in Chaps. 6–12, 14 and 16.

1.2 Thermodynamic Considerations 5

1.2 Thermodynamic Considerations

The stated governing partial differential equations (1.1)–(1.5) for the dy-namics of a compressible material are insufficient to completely describe thephysical processes involved. There are more unknowns than equations andthus closure conditions are required. Physically, such conditions are state-ments related to the nature of the material or medium. Relation (1.6) definesthe total energy E in terms of the velocity vector V involved in equations(1.1)–(1.5) and a new variable e, the specific internal energy. One thereforerequires another relation defining e in terms of quantities already given, suchas pressure and density, as a closure condition. For some applications, or whenmore physical effects are added to the basic equations (1.1)–(1.5), other vari-ables, such as temperature for instance, may need to be introduced.

Central to providing closure conditions is a discussion of the fundamen-tals of Thermodynamics. This introduces new physical variables and providesrelations between variables. Under certain conditions the governing equationsmay be approximated so as to make such discussion of Thermodynamics un-necessary; two important examples are incompressible flows and isentropicflows [112]. The specific internal energy e has an important role in the FirstLaw of Thermodynamics, while the entropy s is intimately involved with theSecond Law of Thermodynamics. Entropy plays a fundamental role not justin establishing the governing equations but also at the level of their mathe-matical properties and the designing of numerical methods to solve them. Inaddition to the basic thermodynamic variables density ρ, pressure p, temper-ature T , specific internal energy e and entropy s, one may define other newvariables that are combinations of these.

1.2.1 Units of Measure

A brief discussion of physical quantities and their units of measure is essen-tial. We consistently adopt, unless otherwise stated, the International Systemof Units or SI Units. Three basic quantities are length (l), mass (m) and time(t). The unit of measure of length is: one metre = 1 m. Submultiples are: onedecimetre = 10−1 m, one centimetre = 10−2 m, one millimetre = 10−3 m.Multiples are: 101 m, 102 m and 103 m = one kilometre. From length one canestablish the units of measure of area: one square metre = 1 m2 and the unitsof measure of volume: one cubic metre = 1 m3. The unit of measure of massis: one kilogram = 1 kg. A useful submultiple is: one gram = 1 g = 10−3 kg.As density is ρ = m/V , where V is the total volume of the system, the unit ofmeasure of density is: one kilogram per cubic metre = 1 kg/m3 = 1 kg m−3.The unit of measure of time is: one second = 1 s. The unit of measure of speedis: one metre per second =1 m/s= 1 m s−1. The unit of measure of accelerationis: one metre per second per second = 1 m/s2= 1 m s−2. The unit of measureof force is: one Newton = 1 N. The Newton N is defined as the force requiredto give a mass of 1 kg an acceleration of 1 m s−2. Newton’s Second Law states

6 1 The Equations of Fluid Dynamics

that force = constant × mass × acceleration. The value of the unit of force isthen chosen so as to make force = 1 when constant = 1. Therefore the unit offorce (N) is: 1 kg m s−2. We now give the unit of measure of pressure. Pressurep is the magnitude of force per unit area and therefore its unit of measure is:one Newton per square metre = 1 N m−2 ≡ 1 Pa : one Pascal. Two commonunits of pressure are 1 bar=105 Pa and 1 atm (atmosphere) = 101 325 Pa. Animportant rule in manipulating physical quantities is dimensional consistency.For example, in the expression ρu2 + p in the momentum equation (1.2), thedimensions of ρu2 must be the same as those of (pressure) p. This is easilyverified.

To introduce the unit of measure of energy we first recall the concept ofWork. Work (W ) is done when a force produces a motion and is measured asW = force × distance moved in the direction of the force. The unit of measureof work is: one Joule = 1 J. One Joule is the work done when the point ofapplication of a force of 1 N moves through a distance of 1 m in the directionof the force. As energy is the capacity to perform work, the unit of measureof energy is also one Joule. The temperature T will be measured in terms ofthe Thermodynamic Scale or the Absolute Scale, in which the unit of measureis: one kelvin = 1 K.

Thermodynamic properties of a system that are proportional to the massm of the system are called extensive properties. Examples are the total energyE and the total volume V of a system. Properties that are independent of mare called intensive properties; examples are temperature T and pressure p.Extensive properties may be converted to their specific (intensive) values bydividing that property by its mass m. For instance, from the total volumeV we obtain the specific volume v = V/m (the reader is warned that v isalso used for velocity component). As ρ = m/V , the specific volume is thereciprocal of density. The units of measure of other quantities will be given asthey are introduced.

1.2.2 Equations of State (EOS)

A system in thermodynamic equilibrium can be completely described bythe basic thermodynamic variables pressure p and specific volume v. A familyof states in thermodynamic equilibrium may be described by a curve in the p–v plane, each characterised by a particular value of a variable temperature T .Systems described by the p–v–T variables are usually called p–v–T systems.There are physical situations that require additional variables. Here we areonly interested in p–v–T systems. In these, one can relate the variables viathe thermal equation of state

T = T (p, v) . (1.14)

Two more possible relations are

p = p(T, v) , v = v(T, p) .

1.2 Thermodynamic Considerations 7

The p–v–T relationship changes from substance to substance. For thermallyideal gases one has the simple expression

T =pv

R, (1.15)

where R is a constant which depends on the particular gas under considera-tion.

The First Law of Thermodynamics states that for a non–adiabatic sys-tem the change Δe in internal energy e in a process will be given byΔe = ΔW + ΔQ, where ΔW is the work done on the system and ΔQ isthe heat transmitted to the system. Taking the work done as dW = −pdv onemay write

dQ = de + pdv . (1.16)

The internal energy e can also be related to p and v via a caloric equation ofstate

e = e(p, v) . (1.17)

Two more possible ways of expressing the p–v–e relationship are

p = p(v, e) , v = v(e, p) .

For a calorically ideal gas one has the simple expression

e =pv

γ − 1=

p

ρ(γ − 1), (1.18)

where γ is a constant that depends on the particular gas under consideration.The thermal and caloric equations of state for a given material are closely

related. Both are necessary for a complete description of the thermodynamicsof a system. Choosing a thermal EOS does restrict the choice of a caloricEOS but does not determine it. Note that for the Euler equations (1.1)–(1.5)one only requires a caloric EOS, e.g. p = p(ρ, e), unless temperature T isneeded for some other purpose, in which case a thermal EOS needs to begiven explicitly.

1.2.3 Other Variables and Relations

The entropy s results as follows. We first introduce an integrating factor1/T so that the expression

de + pdv =(

∂e

∂v+ p

)dv +

∂e

∂pdp

in (1.16) becomes an exact differential. Then the Second Law of Thermody-namics introduces a new variable s, called entropy, via the relation

Tds = de + pdv . (1.19)

8 1 The Equations of Fluid Dynamics

For any process the change in entropy is Δs = Δs0 + Δsi, where Δs0 is theentropy carried into the system through the boundaries of the system andΔsi is the entropy generated in the system during the process. Examples ofentropy–generating mechanisms are heat transfer and viscosity, such as mayoperate within the internal structure of shock waves. The Second Law ofThermodynamics states that Δsi > 0 in any irreversible process. Only in areversible process is Δsi = 0.

Another variable of interest is the specific enthalpy h. This is defined interms of other thermodynamic variables, namely

h = e + pv . (1.20)

One can also establish various relationships amongst the basic thermodynamicvariables already defined. For instance from (1.19)

de = Tds − pdv , (1.21)

that is to say, one may choose to express the internal energy e in terms of thevariables appearing in the differentials, i.e.

e = e(s, v) . (1.22)

Also, taking the differential of (1.20) we have dh = de + pdv + vdp, which byvirtue of (1.21) becomes

dh = Tds + vdp , (1.23)

and thus we can choose to define h in terms of s and p, i.e.

h = h(s, p) . (1.24)

Relations (1.22) and (1.24) are called canonical equations of state and, unlikethe thermal and caloric equations of state (1.14) and (1.17), each of theseprovides a complete description of the Thermodynamics. For instance, given(1.22) in which e is a function of s and v (independent variables) the pressurep and temperature T follow as

p = −(

∂e

∂v

)

s

, T =(

∂e

∂s

)

v

. (1.25)

Relations (1.25) follow from comparing

de =(

∂e

∂s

)

v

ds +(

∂e

∂v

)

s

dv

with equation (1.21). It is conventional in Thermodynamics to specify clearlythe independent variables in partial differentiation, as changes of variablesoften take place. In (1.25), obviously the independent variables are s and v,as is also indicated in (1.22). For instance, the first partial derivative in (1.25)

1.2 Thermodynamic Considerations 9

means differentiation of e with respect to v while holding s constant; thesecond partial derivative in (1.25) means differentiation of e with respect to swhile holding v constant. In a similar manner, equation (1.24) (where s andp are the independent variables) produces T and v from relation (1.23) and

dh =(

∂h

∂s

)

p

ds +(

∂h

∂p

)

s

dp .

Hence,

T =(

∂h

∂s

)

p

, v =(

∂h

∂p

)

s

. (1.26)

The Helmholtz free energy f is defined as

f = e − Ts . (1.27)

A corresponding canonical EOS is

f = f(v, T ) ,

from which one can obtain

s = −(

∂f

∂T

)

v

, p = −(

∂f

∂v

)

T

. (1.28)

Two more quantities can be defined if a thermal EOS v = v(p, T ) is given.These are the volume expansivity α (or expansion coefficient) and the isother-mal compressibility β, namely

α =1v

(∂v

∂T

)

p

, β = −1v

(∂v

∂p

)

T

. (1.29)

Using equations (1.28) and (1.27) we obtain(

∂s

∂v

)

T

=(

∂p

∂T

)

v

=α

β,

from which it can be shown that(

∂e

∂v

)

T

=αT − βp

β. (1.30)

The heat capacity at constant pressure cp and the heat capacity at constantvolume cv (specific heat capacities) are now introduced. In general, when anaddition of heat dQ changes the temperature by dT the ratio c = dQ/dTis called the heat capacity of the system. For a process at constant pressurerelation (1.16) becomes

dQ = de + d(pv) = dh ,