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Advanced numerical methodsfor nonlinear advectiondiffusion-reactionequationsPeter Frolkovič, University of Heidelberg

ContentMotivation and backgroundR 3 TNumerical modellingadvectionadvection + retardation + reactionadvection + nonlinear retardationadvective level set equationKiel, 23.6.20062peter.frolkovic@uni-hd.de

Motivation and BackgroundUG software toolbox - Unstructured Grids“... to simplify the implementation of paralleladaptive multigrid method on unstructuredgrid for complex engineering applications.”P. Bastian et. al. 1997Kiel, 23.6.20063peter.frolkovic@uni-hd.de

Motivation and BackgroundLocally adapted multilevel gridconformingmultilevel grid structurecoarsening possibleKiel, 23.6.20064peter.frolkovic@uni-hd.de

Motivation and BackgroundD 3 F application based on UG (1995-1998)Distributed Density Driven Flownumerical modelling of gravity induced flows near saltdomesFrolkovic, De Schepper: Numerical modeling of convection dominated transport coupled with density driven flow in porous media;Advances in Water Resources, 2001Kiel, 23.6.20065peter.frolkovic@uni-hd.de

Motivation and BackgroundR 3 T application based on UG(1999-2004)Reaction Retardation Radionuclides Transportnumerical modelling of radioactive contaminant transportF., Lampe, Wittum: r3t - software package for numerical simulations of radioactive contaminant transport in groundwater; WiR 2005Kiel, 23.6.20066peter.frolkovic@uni-hd.de

R 3 TRRRT - Radionuclides Reactions (Decay)238238 238Np U PuU233U234decay chains of up to 40 nuclides∂ t C i = ∑ k λki C k − λ ij C iKiel, 23.6.20067peter.frolkovic@uni-hd.de

R 3 TRRRT - TransportNuclides in flowing groundwater238238 238Np U PuU233 234Uconvection-dispersion-diffusion PDEs (up to 40)∂ t C i + ⃗ V · ∇C i − ∇ · D i ( ⃗ V )∇C i = . . .Kiel, 23.6.20068peter.frolkovic@uni-hd.de

R 3 TRRRT - Retardation of transportsorptionNuclides in flowing groundwaterimmobilization238238 238Np U PuU233 234Uup to 120 additional ordinary differential equations∂ t(R i C i) + k i ( K i C i − C i ad)+ . . .Kiel, 23.6.20069peter.frolkovic@uni-hd.de

R 3 TIllustrative example (see video on my homepage)∂ t(R i C i) + k i ( K i C i − C i ad)+ . . .Kiel, 23.6.200610peter.frolkovic@uni-hd.de

R 3 TLinear casesorptionNuclides in flowing groundwaterimmobilization238238 238Np U PuU233 234U∂ t(R i C i) + k i ( K i C i − C i ad)+ . . .Kiel, 23.6.200611peter.frolkovic@uni-hd.de

R 3 TNonlinear casesorptionNuclides in flowing groundwaterimmobilization238238Np U Pu238U233 234U∂ t(R i (C)C i) + k i ( K i (C)C i − C i ad)+ . . .Kiel, 23.6.200612peter.frolkovic@uni-hd.de

R 3 TSparsity of differential equationsU238 238Pu⎛⎝ ∗ 0 00 ∗ 00 0 ∗⎞⎠⎛⎝ ∂ tU 238∂ t P 238∂ t U 234⎞⎠ +U234⎛⎝ T 0 00 T 00 0 T⎞⎠⎛⎝ U 238P 238U 234⎞⎠ +T := ⃗u · ∇ − ∇ · D∇⎛⎝ ∗ 0 00 ∗ 0∗ ∗ ∗⎞⎠⎛⎝ U 238P 238U 234⎞⎠ = 0Kiel, 23.6.200613peter.frolkovic@uni-hd.de

R 3 TSparsity of discrete equations⎛⎜⎝⎞T ii T ij T ik 0 0 0 0 0 0T ji T jj T jk 0 0 0 0 0 0T ki T kj T kk 0 0 0 0 0 00 0 0 T ii T ij T ik 0 0 00 0 0 T ji T jj T jk 0 0 00 0 0 T ki T kj T kk 0 0 00 0 0 0 0 0 T ii T ij T ik⎟0 0 0 0 0 0 T ji T jj T jk ⎠0 0 0 0 0 0 T ki T kj T kk⎛⎜⎝U 238iU 238jU 238kP 238iP 238jP 238kU 234iU 234jU 234k⎞⎟⎠local stiff matrix for a triangle finite elementKiel, 23.6.200614peter.frolkovic@uni-hd.de

R 3 TSparse matrix storage method (Neuss, 1999)Jsparse:Daa="*000*0***";Jsparse:Taa="a000a000a";UU238 238234PuKiel, 23.6.200615peter.frolkovic@uni-hd.de

R 3 T - numerical modellingFinite volume methodsx j"ij!iT exi" eikx kGrid - unstructurednumerical solution given pointwisegradient easily obtained from FE interpolationvertex-centred finite volume method (FVM)finite volume mesh dual to finite elementsKiel, 23.6.200616peter.frolkovic@uni-hd.de

R 3 T - numerical modellingNumerical solutionpiecewise linear, continuousc(t n , x) = c n i + ∇| T e c n · (x − x i ) ,x ∈ T epiecewise constant, discontinuousc(t n , x) = c n i ,x ∈ Ω ipiecewise linear reconstruction, discontinuousc(t n , x) = c n i + ∇| Ω ic n · (x − x i ) ,x ∈ Ω iKiel, 23.6.200617peter.frolkovic@uni-hd.de

Motivation and BackgroundNumerical modellingnumerical algorithms fit analytical modelpreserving 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)Kiel, 23.6.200618peter.frolkovic@uni-hd.de

Advection-Diffusion-DispersionModel equation∂ t c + ∇ · ⃗J = 0 ,⃗ J = ⃗ V c − D∇cKiel, 23.6.200619peter.frolkovic@uni-hd.de

Advection-Diffusion-DispersionModel equation∂ t c + ∇ · ⃗J = 0 ,⃗ J = ⃗ V c − D∇cFVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ J n+1/2ijexact integral formulation:∫Ω ic(t n+1 ) =∫Ω ic(t n ) −t n+1∫t n ∑ ∫∂Ω i ∩∂Ω j⃗n · ⃗JKiel, 23.6.200620peter.frolkovic@uni-hd.de

Advection-Diffusion-DispersionModel equation∂ t c + ∇ · ⃗J = 0 ,⃗ J = ⃗ V c − D∇cFVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ J n+1/2ijphysical property - “mass”|Ω i |c n i :≈ ∫ Ω ic(t n , x) dxKiel, 23.6.200621peter.frolkovic@uni-hd.de

Advection-Diffusion-DispersionModel equation∂ t c + ∇ · ⃗J = 0 ,⃗ J = ⃗ V c − D∇cFVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ J n+1/2ijphysical property - “conservation law”J n+1/2ij= −J n+1/2jiKiel, 23.6.200622peter.frolkovic@uni-hd.de

Advection-Diffusion-DispersionModel equation∂ t c + ∇ · ⃗J = 0 ,⃗ J = ⃗ V c − D∇cFVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ J n+1/2ijfully coupled implicit discretizationJ n+1/2ij = J ij (c n+1i , c n+1j , · · ·)multigrid linear solver, ...Kiel, 23.6.200623peter.frolkovic@uni-hd.de

AdvectionModel equation(⃗V c)∂ t c + ∇ ·= 0Kiel, 23.6.200624peter.frolkovic@uni-hd.de

AdvectionMotivation - exact “simulation” (see video on my homepage)Kiel, 23.6.200625peter.frolkovic@uni-hd.de

AdvectionModel equation(⃗V c)∂ t c + ∇ ·= 0Kiel, 23.6.200626peter.frolkovic@uni-hd.de

AdvectionModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “mass”|Ω i |c n i :≈ ∫ Ω ic(t n , x) dxKiel, 23.6.200628peter.frolkovic@uni-hd.de

AdvectionModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “conservation law”V ij = −V ji ,c n+1/2ij= c n+1/2jiKiel, 23.6.200629peter.frolkovic@uni-hd.de

AdvectionModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “characteristic curves”c n+1/2ij :=?Kiel, 23.6.200630peter.frolkovic@uni-hd.de

Advection - 1st order schemeModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijPiecewise constant numerical solutionc n+1/2ij ={cni V ij > 0c n j V ij < 0Kiel, 23.6.200631peter.frolkovic@uni-hd.de

Advection - 1st order schemeModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “residence time”0 = |Ω i | − τ i∑ max{0, Vij }Kiel, 23.6.200632peter.frolkovic@uni-hd.de

Advection - 1st order schemeModel equation(⃗V c)∂ t c + ∇ ·= 0FVM|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “CFL condition”∆t n ≤ τ iKiel, 23.6.200633peter.frolkovic@uni-hd.de

Advection - 1st order schemeCourant number = 1Courant number > 1Courant number < 1Kiel, 23.6.200634peter.frolkovic@uni-hd.de

Advection - 1st order schemeFlux-based method of characteristics}DistributeMass(j, t 0 , ˆτ, q) {t 0 = t 0 + τ j ;if (t 0 ≥ t n+1 ) then {}b j = b j + ˆτ q ;return;if (t 0 + ˆτ > t n+1 ) then {}j m−1 = j ;b j = b j + (ˆτ − (t n+1 − t 0 )) q ;ˆτ = t n+1 − t 0 ;for (j m−2 ∈ Λ outj m−1)DistributeMass(j m−2 , t 0 , ˆτ, v j m−1 j m−2v jm−1q) ;return ;F.: Flux-based method of characteristics for transport in porous media; CVS, 2002Kiel, 23.6.200635peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationCourant number≈ 5Computation time≈ 2.5 hoursKiel, 23.6.200636peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationCourant number≈ 15Computation time ≈ 1.7 hoursKiel, 23.6.200637peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationExample of 3 radionuclidesR1=1, R2=3, R3=9, small physical dispersionV = (1,0), small dispersion, linear decay chaininitially only 1st component non-zeroKiel, 23.6.200638peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationExample of 3 radionuclidesR1=1, R2=3, R3=9, small physical dispersion2nd order Godunov method with many time stepsKiel, 23.6.200639peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationExample of 3 radionuclidesR1=1, R2=3, R3=9, small physical dispersionstandard operator splitting method, 2 time steps2nd order Godunov method with many time stepsKiel, 23.6.200640peter.frolkovic@uni-hd.de

Advection and Reaction and RetardationExample of 3 radionuclidesR1=1, R2=3, R3=9, small physical dispersionflux-based method of characteristicsF.: Flux-based method of characteristics for coupled system of transport equations in in porous media; CVS, 2002Kiel, 23.6.200641peter.frolkovic@uni-hd.de

AdvectionGodunov methoduse exact solution of related simpler problem1D Riemann’s problemjustified 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)Kiel, 23.6.200642peter.frolkovic@uni-hd.de

Advection and retardationModel equation(⃗V c)∂ t (Rφc) + ∇ ·= 0Fast sorption (equilibrium)R := 1 + 1−φφρKlinear caseR = R(x)Kiel, 23.6.200643peter.frolkovic@uni-hd.de

Advection and retardationExample - Henry isotherm (see video on my homepage)R = 2Kiel, 23.6.200644peter.frolkovic@uni-hd.de

Advection and retardationModel equation(⃗V c)∂ t (Rφc) + ∇ ·= 0Fast sorptionR := 1 + 1−φφρKnonlinear caseR = R(x, c)Kiel, 23.6.200645peter.frolkovic@uni-hd.de

Advection and retardationExample - Freundlich isotherm (see video on my homepage)R = 1 + u p−1Kiel, 23.6.200646peter.frolkovic@uni-hd.de

Advection and retardationNonlinear hyperbolic equation( )⃗V c∂ t θ + ∇ ·= 0θ = θ(c),c = θ −1 (c)1shocks0.8correct speedsharp also with diffusionrarefaction waves0.60.40.2F., Kačur: Semi-analytical solutions of contaminant transport equation withnonlinear sorption in 1D; Comp. Geosciences, 2006, to appear00.5 1 1.5 2 2.5 3 3.5xKiel, 23.6.200647peter.frolkovic@uni-hd.de

Advection and retardationImplementation(see video on my homepage)Kiel, 23.6.2006linear sorption48nonlinear sorptionpeter.frolkovic@uni-hd.de

Advection - 1st order methodTrivial example∂ t c + ⃗ V · ∇c = 0 ,⃗ V · ∇c(0, x) ≡ constc n+1i= c n i − ∆t n constKiel, 23.6.200649peter.frolkovic@uni-hd.de

Advection - 1st order methodConsistent for structured grid?∂ t c + ⃗ V · ∇c = 0 ,⃗ V · ∇c(0, x) ≡ const|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijKiel, 23.6.200650peter.frolkovic@uni-hd.de

Advection - 1st order methodConsistent for structured grid!∂ t c + ⃗ V · ∇c = 0 ,⃗ V · ∇c(0, x) ≡ const|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijKiel, 23.6.200651peter.frolkovic@uni-hd.de

Advection - 1st order methodNonconsistent for unstructured grid!∂ t c + ⃗ V · ∇c = 0 ,⃗ V · ∇c(0, x) ≡ const|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijKiel, 23.6.200652peter.frolkovic@uni-hd.de

Advection - 1st order method (200x200)1t = 01t = 0.196351t = 0.39270.50.50.500 0.5 100 0.5 100 0.5 11t = 0.589051t = 0.78541t = 0.981750.50.50.500 0.5 11t = 1.178100 0.5 11t = 1.374400 0.5 11t = 1.57080.50.50.500 0.5 100 0.5 100 0.5 1Kiel, 23.6.200653peter.frolkovic@uni-hd.de

AdvectionLevel set equation∂ t c + ⃗ V · ∇c = 0 , ∇ · ⃗V = 0Kiel, 23.6.200654peter.frolkovic@uni-hd.de

AdvectionLevel set equationFVM∂ t c + ⃗ V · ∇c = 0 , ∇ · ⃗V = 0|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijKiel, 23.6.200655peter.frolkovic@uni-hd.de

AdvectionLevel set equationFVM∂ t c + ⃗ V · ∇c = 0 , ∇ · ⃗V = 0|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “value”c n i :≈ c(t n , x i )Kiel, 23.6.200656peter.frolkovic@uni-hd.de

AdvectionLevel set equationFVM∂ t c + ⃗ V · ∇c = 0 , ∇ · ⃗V = 0|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “characteristic curves”c n+1/2ij := c(t n , X ij (t n ))Kiel, 23.6.200657peter.frolkovic@uni-hd.de

Advection - 2nd order schemeLevel set equationFVM∂ t c + ⃗ V · ∇c = 0 , ∇ · ⃗V = 0|Ω i |c n+1i= |Ω i |c n i − ∆t n ∑ V ij c n+1/2ijphysical property - “characteristic curves”c n+1/2ij := c(t n , X ij (t n ))Kiel, 23.6.2006c n+1/2ij:= c n ij − ∆tn2 ⃗ V i · ∇c n i58peter.frolkovic@uni-hd.de

Advection - 1st versus 2nd order method10.9t = 010.80.70,80.60,60.5Y0.40,40.30.20,20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 0,2 0,4 0,6 0,8X1Kiel, 23.6.200659peter.frolkovic@uni-hd.de

Advection - 1st order method (200x200)1t = 01t = 0.196351t = 0.39270.50.50.500 0.5 100 0.5 100 0.5 11t = 0.589051t = 0.78541t = 0.981750.50.50.500 0.5 11t = 1.178100 0.5 11t = 1.374400 0.5 11t = 1.57080.50.50.500 0.5 100 0.5 100 0.5 1Kiel, 23.6.200660peter.frolkovic@uni-hd.de

Advection - 2nd order method (200x200)1t = 01t = 0.196351t = 0.39270.50.50.500 0.5 100 0.5 100 0.5 11t = 0.589051t = 0.78541t = 0.981750.50.50.500 0.5 11t = 1.178100 0.5 11t = 1.374400 0.5 11t = 1.57080.50.50.500 0.5 100 0.5 100 0.5 1Kiel, 23.6.200661peter.frolkovic@uni-hd.de

Advection - 2nd order method (200x200)1t = 01t = 0.39271t = 0.78540.50.50.500 0.5 100 0.5 100 0.5 11t = 1.17811t = 1.57081t = 1.96350.50.50.500 0.5 11t = 2.356200 0.5 11t = 2.748900 0.5 11t = 3.14160.50.50.500 0.5 100 0.5 100 0.5 1Kiel, 23.6.200662peter.frolkovic@uni-hd.de

AdvectionFlux-based level set method (see video on my homepage)F., Mikula: High resolution flux-based level set method; 2005Kiel, 23.6.200663peter.frolkovic@uni-hd.de

Nonlinear advective level set equationExample with topological changes (see video on my homepage)F., Mikula: Flux-based level set method: finite volume emthod for evolving interfaces; 2002Kiel, 23.6.200664peter.frolkovic@uni-hd.de

ConclusionsNumerical modellingnumerical algorithms fit analytical modelpreserving physical properties, ...stable, consistent, ...available, simple and good in generalunstructured gridsrobust for rough data, ... (1st order schemes)precise for smooth parts, ... (2nd order schemes)Kiel, 23.6.200665peter.frolkovic@uni-hd.de