Advanced numerical methods for nonlinear advectiondiffusion-reaction. Peter Frolkovič, University of Heidelberg

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1 Advanced numerical methods for nonlinear advectiondiffusion-reaction equations Peter Frolkovič, University of Heidelberg

2 Content Motivation and background R 3 T Numerical modelling advection advection + retardation + reaction advection + nonlinear retardation advective level set equation 2

3 Motivation and Background UG software toolbox - Unstructured Grids... to simplify the implementation of parallel adaptive multigrid method on unstructured grid for complex engineering applications. P. Bastian et. al

4 Motivation and Background Locally adapted multilevel grid conforming multilevel grid structure coarsening possible 4

5 Motivation and Background D 3 F application based on UG ( ) Distributed Density Driven Flow numerical 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, 2 5

6 Motivation and Background R 3 T application based on UG(999-24) Reaction Retardation Radionuclides Transport numerical modelling of radioactive contaminant transport F., Lampe, Wittum: r3t - software package for numerical simulations of radioactive contaminant transport in groundwater; WiR 25 6

7 R 3 T RRRT - Radionuclides Reactions (Decay) Np U Pu U 233 U 234 decay chains of up to 4 nuclides t C i = k λki C k λ ij C i 7

8 R 3 T RRRT - Transport Nuclides in flowing groundwater Np U Pu U U convection-dispersion-diffusion PDEs (up to 4) t C i + V C i D i ( V ) C i =... 8

9 R 3 T RRRT - Retardation of transport sorption Nuclides in flowing groundwater immobilization Np U Pu U U up to 2 additional ordinary differential equations t ( R i C i) + k i ( K i C i C i ad)

10 R 3 T Illustrative example (see video on my homepage) t ( R i C i) + k i ( K i C i C i ad) +...

11 R 3 T Linear case sorption Nuclides in flowing groundwater immobilization Np U Pu U U t ( R i C i) + k i ( K i C i C i ad) +...

12 R 3 T Nonlinear case sorption Nuclides in flowing groundwater immobilization Np U Pu 238 U U t ( R i (C)C i) + k i ( K i (C)C i C i ad)

13 R 3 T Sparsity of differential equations U Pu tu 238 t P 238 t U U 234 T T T U 238 P 238 U T := u D U 238 P 238 U 234 = 3

14 R 3 T Sparsity of discrete equations T ii T ij T ik T ji T jj T jk T ki T kj T kk T ii T ij T ik T ji T jj T jk T ki T kj T kk T ii T ij T ik T ji T jj T jk T ki T kj T kk U 238 i U 238 j U 238 k P 238 i P 238 j P 238 k U 234 i U 234 j U 234 k local stiff matrix for a triangle finite element 4

15 R 3 T Sparse matrix storage method (Neuss, 999) Jsparse:Daa=" * * ***"; Jsparse:Taa=" a a a"; U U Pu 5

16 R 3 T - numerical modelling Finite volume methods x j " ij! i T e x i " e ik x k Grid - unstructured numerical solution given pointwise gradient easily obtained from FE interpolation vertex-centred finite volume method (FVM) finite volume mesh dual to finite elements 6

17 R 3 T - numerical modelling Numerical solution piecewise linear, continuous c(t n, x) = c n i + T e c n (x x i ), x T e piecewise constant, discontinuous c(t n, x) = c n i, x Ω i piecewise linear reconstruction, discontinuous c(t n, x) = c n i + Ω i c n (x x i ), x Ω i 7

18 Motivation and Background Numerical modelling numerical algorithms fit analytical model preserving physical properties,... stable, consistent,... available, simple and good in general: unstructured grids robust for rough data,... (st order schemes) precise for smooth parts,... (2nd order schemes) 8

19 Advection-Diffusion-Dispersion Model equation t c + J =, J = V c D c 9

20 Advection-Diffusion-Dispersion Model equation t c + J =, J = V c D c FVM Ω i c n+ i = Ω i c n i t n J n+/2 ij exact integral formulation: t n+ Ω i c(t n+ ) = Ω i c(t n ) t n Ω i Ω j n J 2

21 Advection-Diffusion-Dispersion Model equation t c + J =, J = V c D c FVM Ω i c n+ i = Ω i c n i t n J n+/2 ij physical property - mass Ω i c n i : Ω i c(t n, x) dx 2

22 Advection-Diffusion-Dispersion Model equation t c + J =, J = V c D c FVM Ω i c n+ i = Ω i c n i t n J n+/2 ij physical property - conservation law J n+/2 ij = J n+/2 ji 22

23 Advection-Diffusion-Dispersion Model equation t c + J =, J = V c D c FVM Ω i c n+ i = Ω i c n i t n J n+/2 ij fully coupled implicit discretization J n+/2 ij = J ij (c n+ i, c n+ j, ) multigrid linear solver,... 23

24 Advection Model equation ( V c ) t c + = 24

25 Advection Motivation - exact simulation (see video on my homepage) 25

26 Advection Model equation ( V c ) t c + = 26

27 Advection Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij 27

28 Advection Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - mass Ω i c n i : Ω i c(t n, x) dx 28

29 Advection Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - conservation law V ij = V ji, c n+/2 ij = c n+/2 ji 29

30 Advection Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - characteristic curves c n+/2 ij :=? 3

31 Advection - st order scheme Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij Piecewise constant numerical solution { c n i V ij > c n+/2 ij = c n j V ij < 3

32 Advection - st order scheme Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - residence time = Ω i τ i max{, Vij } 32

33 Advection - st order scheme Model equation ( V c ) t c + = FVM Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - CFL condition t n τ i 33

34 Advection - st order scheme Courant number = Courant number > Courant number < 34

35 Advection - st order scheme Flux-based method of characteristics } DistributeMass(j, t, ˆτ, q) { t = t + τ j ; if (t t n+ ) then { } b j = b j + ˆτ q ; return; if (t + ˆτ > t n+ ) then { } j m = j ; b j = b j + (ˆτ (t n+ t )) q ; ˆτ = t n+ t ; for (j m 2 Λ out j m ) DistributeMass(j m 2, t, ˆτ, v j m j m 2 v jm q) ; return ; F.: Flux-based method of characteristics for transport in porous media; CVS, 22 35

36 Advection and Reaction and Retardation Courant number 5 Computation time 2.5 hours 36

37 Advection and Reaction and Retardation Courant number 5 Computation time.7 hours 37

38 Advection and Reaction and Retardation Example of 3 radionuclides R=, R2=3, R3=9, small physical dispersion V = (,), small dispersion, linear decay chain initially only st component non-zero 38

39 Advection and Reaction and Retardation Example of 3 radionuclides R=, R2=3, R3=9, small physical dispersion 2nd order Godunov method with many time steps 39

40 Advection and Reaction and Retardation Example of 3 radionuclides R=, R2=3, R3=9, small physical dispersion standard operator splitting method, 2 time steps 2nd order Godunov method with many time steps 4

41 Advection and Reaction and Retardation Example of 3 radionuclides R=, R2=3, R3=9, small physical dispersion flux-based method of characteristics F.: Flux-based method of characteristics for coupled system of transport equations in in porous media; CVS, 22 4

42 Advection Godunov method use exact solution of related simpler problem D Riemann s problem justified by numerical hyperbolic equations e.g., D advection => st order upwind m. High-resolution FVM piecewise linear numerical solution structured grid - Leveque 22 unstructured grid? (e.g., Sonar 993) 42

43 Advection and retardation Model equation ( V c ) t (Rφc) + = Fast sorption (equilibrium) R := + φ φ ρk linear case R = R(x) 43

44 Advection and retardation Example - Henry isotherm (see video on my homepage) R = 2 44

45 Advection and retardation Model equation ( V c ) t (Rφc) + = Fast sorption R := + φ φ ρk nonlinear case R = R(x, c) 45

46 Advection and retardation Example - Freundlich isotherm (see video on my homepage) R = + u p 46

47 Advection and retardation Nonlinear hyperbolic equation ( ) V c t θ + = θ = θ(c), c = θ (c) shocks.8 correct speed sharp also with diffusion rarefaction waves F., Kačur: Semi-analytical solutions of contaminant transport equation with nonlinear sorption in D; Comp. Geosciences, 26, to appear x 47

48 Advection and retardation Implementation (see video on my homepage) linear sorption 48 nonlinear sorption

49 Advection - st order method Trivial example t c + V c =, V c(, x) const c n+ i = c n i t n const 49

50 Advection - st order method Consistent for structured grid? t c + V c =, V c(, x) const Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij 5

51 Advection - st order method Consistent for structured grid! t c + V c =, V c(, x) const Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij 5

52 Advection - st order method Nonconsistent for unstructured grid! t c + V c =, V c(, x) const Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij 52

53 Advection - st order method (2x2) t = t =.9635 t = t =.5895 t =.7854 t = t =.78.5 t = t =

54 Advection Level set equation t c + V c =, V = 54

55 Advection Level set equation FVM t c + V c =, V = Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij 55

56 Advection Level set equation FVM t c + V c =, V = Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - value c n i : c(t n, x i ) 56

57 Advection Level set equation FVM t c + V c =, V = Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - characteristic curves c n+/2 ij := c(t n, X ij (t n )) 57

58 Advection - 2nd order scheme Level set equation FVM t c + V c =, V = Ω i c n+ i = Ω i c n i t n V ij c n+/2 ij physical property - characteristic curves c n+/2 ij := c(t n, X ij (t n )) c n+/2 ij := c n ij tn 2 V i c n i 58

59 Advection - st versus 2nd order method.9 t =.8.7,8.6,6.5 Y.4,4.3.2, ,2,4,6,8 X 59

60 Advection - st order method (2x2) t = t =.9635 t = t =.5895 t =.7854 t = t =.78.5 t = t =

61 Advection - 2nd order method (2x2) t = t =.9635 t = t =.5895 t =.7854 t = t =.78.5 t = t =

62 Advection - 2nd order method (2x2) t = t =.3927 t = t =.78 t =.578 t = t = t = t =

63 Advection Flux-based level set method (see video on my homepage) F., Mikula: High resolution flux-based level set method; 25 63

64 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; 22 64

65 Conclusions Numerical modelling numerical algorithms fit analytical model preserving physical properties,... stable, consistent,... available, simple and good in general unstructured grids robust for rough data,... (st order schemes) precise for smooth parts,... (2nd order schemes) 65