Advanced numerical methods for nonlinear advection- …pfrolkovic/kiel.pdffor nonlinear...

Advanced numerical methodsfor nonlinear advection-

diffusion-reaction equationsPeter Frolkovič, University of Heidelberg

2Kiel, 23.6.2006 [email protected]

Content

Motivation and background R3T

Numerical modellingadvectionadvection + retardation + reaction advection + nonlinear retardationadvective level set equation


Motivation and BackgroundUG software toolbox - Unstructured Grids

“... to simplify the implementation of parallel adaptive multigrid method on unstructured grid for complex engineering applications.”

P. Bastian et. al. 1997


Motivation and BackgroundLocally adapted multilevel grid

conformingmultilevel grid structure

coarsening possible


Motivation and Background

D3F application based on UG (1995-1998)

Distributed Density Driven Flownumerical modelling of gravity induced flows near saltdomes

Frolkovic, De Schepper: Numerical modeling of convection dominated transport coupled with density driven flow in porous media;Advances in Water Resources, 2001



R3T application based on UG(1999-2004)

Reaction Retardation Radionuclides Transportnumerical modelling of radioactive contaminant transport

F., Lampe, Wittum: r3t - software package for numerical simulations of radioactive contaminant transport in groundwater; WiR 2005

[email protected], 23.6.2006

R3T

234

Np U Pu238

U U

238 238

233

RRRT - Radionuclides Reactions (Decay)

∂tCi=

∑k λkiCk

− λijCi

decay chains of up to 40 nuclides


R3TRRRT - Transport

Nuclides in flowing groundwater

Np U Pu238

U U

238 238

233 234

convection-dispersion-diffusion PDEs (up to 40)

∂tCi + "V ·∇Ci − ∇ · Di("V )∇Ci = . . .


R3TRRRT - Retardation of transport


Np U Pu

immobilizationsorption

238

U U

238 238

233 234

up to 120 additional ordinary differential equations

∂t

(RiCi

)+ ki

(KiCi − Ci

ad

)+ . . .


R3TIllustrative example (see video on my homepage)

∂t

(RiCi

)+ ki

(KiCi − Ci

ad

)+ . . .


R3TLinear case


Np U Pu


238

U U

238 238

233 234

∂t

(RiCi

)+ ki

(KiCi

− Ciad

)+ . . .


R3T


Np U Pu


U U

238

233 234

238238

Nonlinear case

∂t

(Ri(C)Ci

)+ ki

(Ki(C)Ci

− Ciad

)+ . . .


R3T

∗ 0 0

0 ∗ 0

0 0 ∗

∂tU238

∂tP238

∂tU234

+

T 0 0

0 T 0

0 0 T

U238

P 238

U234

+

∗ 0 0

0 ∗ 0

∗ ∗ ∗

U238

P 238

U234

= 0

T := !u ·∇−∇ · D∇

234

U Pu

U

238 238

Sparsity of differential equations


R3T

Tii Tij Tik 0 0 0 0 0 0

Tji Tjj Tjk 0 0 0 0 0 0

Tki Tkj Tkk 0 0 0 0 0 0

0 0 0 Tii Tij Tik 0 0 0

0 0 0 Tji Tjj Tjk 0 0 0

0 0 0 Tki Tkj Tkk 0 0 0

0 0 0 0 0 0 Tii Tij Tik

0 0 0 0 0 0 Tji Tjj Tjk

0 0 0 0 0 0 Tki Tkj Tkk

U238i

U238j

U238k

P 238i

P 238j

P 238k

U234i

U234j

U234k

Sparsity of discrete equations

local stiff matrix for a triangle finite element


R3T

Jsparse:Daa=" *00 0*0 ***";Jsparse:Taa=" a00 0a0 00a";

234

U Pu

U

238 238

Sparse matrix storage method (Neuss, 1999)


R3T - numerical modellingFinite volume methods

Grid - unstructurednumerical solution given pointwisegradient easily obtained from FE interpolation

vertex-centred finite volume method (FVM)finite volume mesh dual to finite elements

i

xx k

ijxj

i

T e

!"

"e

ik


R3T - numerical modellingNumerical solution

piecewise linear, continuous

piecewise constant, discontinuous

piecewise linear reconstruction, discontinuousc(tn, x) = cn

i, x ∈ Ωi

c(tn, x) = cni + ∇|T ecn · (x − xi) , x ∈ T e

c(tn, x) = cni

+ ∇|Ωicn · (x − xi) , x ∈ Ωi



Numerical modellingnumerical algorithms fit analytical model

preserving physical properties, ...stable, consistent, ...

available, simple and good in general:unstructured gridsrobust for rough data, ... (1st order schemes)

precise for smooth parts, ... (2nd order schemes)


Advection-Diffusion-DispersionModel equation

∂tc +∇ · "J = 0 , "J = "V c−D∇c



FVM

exact integral formulation:

∂tc +∇ · "J = 0 , "J = "V c−D∇c

|Ωi|cn+1i = |Ωi|cn

i −∆tn∑

Jn+1/2ij

∫Ωi

c(tn+1) =∫

Ωi

c(tn)−tn+1∫tn

∑ ∫∂Ωi∩∂Ωj

!n · !J



FVM

physical property - “mass”

∂tc +∇ · "J = 0 , "J = "V c−D∇c


i −∆tn∑

Jn+1/2ij

|Ωi|cni :≈ ∫

Ωic(tn, x) dx



FVM

physical property - “conservation law”

∂tc +∇ · "J = 0 , "J = "V c−D∇c


i −∆tn∑

Jn+1/2ij

Jn+1/2ij = −Jn+1/2

ji



FVM

fully coupled implicit discretization

multigrid linear solver, ...

∂tc +∇ · "J = 0 , "J = "V c−D∇c


i −∆tn∑

Jn+1/2ij

Jn+1/2ij = Jij(cn+1

i , cn+1j , · · ·)


AdvectionModel equation

∂tc + ∇ ·

("V c

)= 0


AdvectionMotivation - exact “simulation” (see video on my homepage)



∂tc + ∇ ·

("V c

)= 0



FVM∂tc + ∇ ·

("V c

)= 0


i −∆tn∑

Vijcn+1/2ij



FVM

physical property - “mass”

∂tc + ∇ ·

("V c

)= 0


i −∆tn∑

Vijcn+1/2ij

|Ωi|cni :≈ ∫

Ωic(tn, x) dx



FVM

physical property - “conservation law”

∂tc + ∇ ·

("V c

)= 0

Vij = −Vji , cn+1/2ij = cn+1/2

ji


i −∆tn∑

Vijcn+1/2ij



FVM

physical property - “characteristic curves”

cn+1/2ij :=?


i −∆tn∑

Vijcn+1/2ij

∂tc + ∇ ·

("V c

)= 0


Advection - 1st order schemeModel equation

FVM

Piecewise constant numerical solution

cn+1/2ij =

cni Vij > 0

cnj Vij < 0


i −∆tn∑

Vijcn+1/2ij

∂tc + ∇ ·

("V c

)= 0



FVM

physical property - “residence time”

∂tc + ∇ ·

("V c

)= 0


i −∆tn∑

Vijcn+1/2ij

0 = |Ωi|− τi∑

max0, Vij



FVM

physical property - “CFL condition”

∂tc + ∇ ·

("V c

)= 0


i −∆tn∑

Vijcn+1/2ij

∆tn ≤ τi


Advection - 1st order schemeCourant number = 1

Courant number > 1

Courant number < 1


Advection - 1st order schemeFlux-based method of characteristicsDistributeMass(j, t0, τ , q)

t0 = t0 + τj ;if (t0 ≥ tn+1) then

bj = bj + τ q ;return;

if (t0 + τ > tn+1) then

bj = bj + (τ − (tn+1 − t0)) q ;τ = tn+1 − t0 ;

jm−1 = j ;for (jm−2 ∈ Λout

jm−1)

DistributeMass(jm−2, t0, τ ,vjm−1jm−2

vjm−1q) ;

return ;

F.: Flux-based method of characteristics for transport in porous media; CVS, 2002


Advection and Reaction and Retardation

Courant number

Computation time

≈ 5

≈ 2.5 hours


Advection and Reaction and Retardation

Courant number

Computation time ≈ 1.7 hours

≈ 15


Advection and Reaction and RetardationExample of 3 radionuclides

R1=1, R2=3, R3=9, small physical dispersion

V = (1,0), small dispersion, linear decay chaininitially only 1st component non-zero




2nd order Godunov method with many time steps




standard operator splitting method, 2 time steps

2nd order Godunov method with many time steps




flux-based method of characteristics

F.: Flux-based method of characteristics for coupled system of transport equations in in porous media; CVS, 2002


AdvectionGodunov method

use exact solution of related simpler problem1D Riemann’s problem

justified by numerical hyperbolic equationse.g., 1D advection => 1st order upwind m.

High-resolution FVMpiecewise linear numerical solutionstructured grid - Leveque 2002unstructured grid? (e.g., Sonar 1993)


Advection and retardationModel equation

Fast sorption (equilibrium)

linear case

R := 1 + 1−φφ ρK

R = R(x)

∂t (Rφc) +∇ ·(

#V c)

= 0


Advection and retardationExample - Henry isotherm (see video on my homepage)

R = 2


Advection and retardationModel equation

Fast sorption

nonlinear case

R := 1 + 1−φφ ρK

R = R(x, c)

∂t (Rφc) +∇ ·(

#V c)

= 0


Advection and retardationExample - Freundlich isotherm (see video on my homepage)

R = 1 + up−1


Advection and retardationNonlinear hyperbolic equation

shocks

correct speedsharp also with diffusion

rarefaction waves0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5

x

∂tθ + ∇ ·

(#V c

)= 0

θ = θ(c), c = θ−1(c)

F., Kačur: Semi-analytical solutions of contaminant transport equation with nonlinear sorption in 1D; Comp. Geosciences, 2006, to appear


Advection and retardationImplementation (see video on my homepage)

linear sorption nonlinear sorption


Advection - 1st order methodTrivial example∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const

cn+1i = cn

i −∆tnconst


Advection - 1st order methodConsistent for structured grid?∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const


i −∆tn∑

Vijcn+1/2ij


Advection - 1st order methodConsistent for structured grid!∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const


i −∆tn∑

Vijcn+1/2ij


Advection - 1st order methodNonconsistent for unstructured grid!∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const


i −∆tn∑

Vijcn+1/2ij


Advection - 1st order method (200x200)t = 0

0 0.5 10

0.5

1

t = 0.19635

0 0.5 10

0.5

1t = 0.3927

0 0.5 10

0.5

1

t = 0.58905

0 0.5 10

0.5

1t = 0.7854

0 0.5 10

0.5

1t = 0.98175

0 0.5 10

0.5

1

t = 1.1781

0 0.5 10

0.5

1t = 1.3744

0 0.5 10

0.5

1t = 1.5708

0 0.5 10

0.5

1


AdvectionLevel set equation

∂tc + "V ·∇c = 0 , ∇ · "V = 0



FVM


i −∆tn∑

Vijcn+1/2ij

∂tc + "V ·∇c = 0 , ∇ · "V = 0



FVM

physical property - “value”


i −∆tn∑

Vijcn+1/2ij

cni :≈ c(tn, xi)

∂tc + "V ·∇c = 0 , ∇ · "V = 0

∂tc + "V ·∇c = 0 , ∇ · "V = 0



FVM



i −∆tn∑

Vijcn+1/2ij

cn+1/2ij := c(tn,Xij(tn))

∂tc + "V ·∇c = 0 , ∇ · "V = 0


Advection - 2nd order schemeLevel set equation

FVM



i −∆tn∑

Vijcn+1/2ij

cn+1/2ij := c(tn,Xij(tn))

cn+1/2ij := cn

ij − ∆tn

2!Vi ·∇cn

i


Advection - 1st versus 2nd order method

0,20 0,8

X

Y

0,4

0,6

0,8

0

1

0,2

10,60,4

t = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Advection - 1st order method (200x200)t = 0

0 0.5 10

0.5

1

t = 0.19635

0 0.5 10

0.5

1t = 0.3927

0 0.5 10

0.5

1

t = 0.58905

0 0.5 10

0.5

1t = 0.7854

0 0.5 10

0.5

1t = 0.98175

0 0.5 10

0.5

1

t = 1.1781

0 0.5 10

0.5

1t = 1.3744

0 0.5 10

0.5

1t = 1.5708

0 0.5 10

0.5

1


Advection - 2nd order method (200x200)t = 0

0 0.5 10

0.5

1

t = 0.19635

0 0.5 10

0.5

1t = 0.3927

0 0.5 10

0.5

1

t = 0.58905

0 0.5 10

0.5

1t = 0.7854

0 0.5 10

0.5

1t = 0.98175

0 0.5 10

0.5

1

t = 1.1781

0 0.5 10

0.5

1t = 1.3744

0 0.5 10

0.5

1t = 1.5708

0 0.5 10

0.5

1


Advection - 2nd order method (200x200)t = 0

0 0.5 10

0.5

1

t = 0.3927

0 0.5 10

0.5

1t = 0.7854

0 0.5 10

0.5

1

t = 1.1781

0 0.5 10

0.5

1t = 1.5708

0 0.5 10

0.5

1t = 1.9635

0 0.5 10

0.5

1

t = 2.3562

0 0.5 10

0.5

1t = 2.7489

0 0.5 10

0.5

1t = 3.1416

0 0.5 10

0.5

1


Advection

Flux-based level set method (see video on my homepage)

F., Mikula: High resolution flux-based level set method; 2005


Nonlinear advective level set equation

Example with topological changes (see video on my homepage)

F., Mikula: Flux-based level set method: finite volume emthod for evolving interfaces; 2002


ConclusionsNumerical modelling

numerical algorithms fit analytical model preserving physical properties, ...stable, consistent, ...

available, simple and good in generalunstructured gridsrobust for rough data, ... (1st order schemes)

precise for smooth parts, ... (2nd order schemes)

Advanced numerical methods for nonlinear advection- …pfrolkovic/kiel.pdffor nonlinear...

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