Post on 24-Feb-2021
Advanced numerical methodsfor nonlinear advection-
diffusion-reaction equationsPeter Frolkovič, University of Heidelberg
2Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Content
Motivation and background R3T
Numerical modellingadvectionadvection + retardation + reaction advection + nonlinear retardationadvective level set equation
3Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
4Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Motivation and BackgroundLocally adapted multilevel grid
conformingmultilevel grid structure
coarsening possible
5Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
6Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Motivation and Background
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
peter.frolkovic@uni-hd.de7Kiel, 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
peter.frolkovic@uni-hd.de8Kiel, 23.6.2006
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 = . . .
peter.frolkovic@uni-hd.de9Kiel, 23.6.2006
R3TRRRT - Retardation of transport
Nuclides in flowing groundwater
Np U Pu
immobilizationsorption
238
U U
238 238
233 234
up to 120 additional ordinary differential equations
∂t
(RiCi
)+ ki
(KiCi − Ci
ad
)+ . . .
10Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
R3TIllustrative example (see video on my homepage)
∂t
(RiCi
)+ ki
(KiCi − Ci
ad
)+ . . .
11Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
R3TLinear case
Nuclides in flowing groundwater
Np U Pu
immobilizationsorption
238
U U
238 238
233 234
∂t
(RiCi
)+ ki
(KiCi
− Ciad
)+ . . .
12Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
R3T
Nuclides in flowing groundwater
Np U Pu
immobilizationsorption
U U
238
233 234
238238
Nonlinear case
∂t
(Ri(C)Ci
)+ ki
(Ki(C)Ci
− Ciad
)+ . . .
13Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
14Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
15Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
R3T
Jsparse:Daa=" *00 0*0 ***";Jsparse:Taa=" a00 0a0 00a";
234
U Pu
U
238 238
Sparse matrix storage method (Neuss, 1999)
16Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
17Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
18Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Motivation and Background
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)
19Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection-Diffusion-DispersionModel equation
∂tc +∇ · "J = 0 , "J = "V c−D∇c
20Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection-Diffusion-DispersionModel equation
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
21Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection-Diffusion-DispersionModel equation
FVM
physical property - “mass”
∂tc +∇ · "J = 0 , "J = "V c−D∇c
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Jn+1/2ij
|Ωi|cni :≈ ∫
Ωic(tn, x) dx
22Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection-Diffusion-DispersionModel equation
FVM
physical property - “conservation law”
∂tc +∇ · "J = 0 , "J = "V c−D∇c
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Jn+1/2ij
Jn+1/2ij = −Jn+1/2
ji
23Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection-Diffusion-DispersionModel equation
FVM
fully coupled implicit discretization
multigrid linear solver, ...
∂tc +∇ · "J = 0 , "J = "V c−D∇c
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Jn+1/2ij
Jn+1/2ij = Jij(cn+1
i , cn+1j , · · ·)
24Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
∂tc + ∇ ·
("V c
)= 0
25Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionMotivation - exact “simulation” (see video on my homepage)
26Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
∂tc + ∇ ·
("V c
)= 0
27Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
FVM∂tc + ∇ ·
("V c
)= 0
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
28Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
FVM
physical property - “mass”
∂tc + ∇ ·
("V c
)= 0
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
|Ωi|cni :≈ ∫
Ωic(tn, x) dx
29Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
FVM
physical property - “conservation law”
∂tc + ∇ ·
("V c
)= 0
Vij = −Vji , cn+1/2ij = cn+1/2
ji
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
30Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionModel equation
FVM
physical property - “characteristic curves”
cn+1/2ij :=?
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
∂tc + ∇ ·
("V c
)= 0
31Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order schemeModel equation
FVM
Piecewise constant numerical solution
cn+1/2ij =
cni Vij > 0
cnj Vij < 0
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
∂tc + ∇ ·
("V c
)= 0
32Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order schemeModel equation
FVM
physical property - “residence time”
∂tc + ∇ ·
("V c
)= 0
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
0 = |Ωi|− τi∑
max0, Vij
33Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order schemeModel equation
FVM
physical property - “CFL condition”
∂tc + ∇ ·
("V c
)= 0
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
∆tn ≤ τi
34Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order schemeCourant number = 1
Courant number > 1
Courant number < 1
35Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
36Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and Reaction and Retardation
Courant number
Computation time
≈ 5
≈ 2.5 hours
37Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and Reaction and Retardation
Courant number
Computation time ≈ 1.7 hours
≈ 15
38Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
39Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and Reaction and RetardationExample of 3 radionuclides
R1=1, R2=3, R3=9, small physical dispersion
2nd order Godunov method with many time steps
40Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and Reaction and RetardationExample of 3 radionuclides
R1=1, R2=3, R3=9, small physical dispersion
standard operator splitting method, 2 time steps
2nd order Godunov method with many time steps
41Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and Reaction and RetardationExample of 3 radionuclides
R1=1, 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, 2002
42Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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)
43Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and retardationModel equation
Fast sorption (equilibrium)
linear case
R := 1 + 1−φφ ρK
R = R(x)
∂t (Rφc) +∇ ·(
#V c)
= 0
44Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and retardationExample - Henry isotherm (see video on my homepage)
R = 2
45Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and retardationModel equation
Fast sorption
nonlinear case
R := 1 + 1−φφ ρK
R = R(x, c)
∂t (Rφc) +∇ ·(
#V c)
= 0
46Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and retardationExample - Freundlich isotherm (see video on my homepage)
R = 1 + up−1
47Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
48Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection and retardationImplementation (see video on my homepage)
linear sorption nonlinear sorption
49Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order methodTrivial example∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const
cn+1i = cn
i −∆tnconst
50Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order methodConsistent for structured grid?∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
51Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order methodConsistent for structured grid!∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
52Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 1st order methodNonconsistent for unstructured grid!∂tc + "V ·∇c = 0 , "V ·∇c(0, x) ≡ const
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
53Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
54Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionLevel set equation
∂tc + "V ·∇c = 0 , ∇ · "V = 0
55Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionLevel set equation
FVM
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
∂tc + "V ·∇c = 0 , ∇ · "V = 0
56Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionLevel set equation
FVM
physical property - “value”
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
cni :≈ c(tn, xi)
∂tc + "V ·∇c = 0 , ∇ · "V = 0
∂tc + "V ·∇c = 0 , ∇ · "V = 0
57Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
AdvectionLevel set equation
FVM
physical property - “characteristic curves”
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
cn+1/2ij := c(tn,Xij(tn))
∂tc + "V ·∇c = 0 , ∇ · "V = 0
58Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection - 2nd order schemeLevel set equation
FVM
physical property - “characteristic curves”
|Ωi|cn+1i = |Ωi|cn
i −∆tn∑
Vijcn+1/2ij
cn+1/2ij := c(tn,Xij(tn))
cn+1/2ij := cn
ij − ∆tn
2!Vi ·∇cn
i
59Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
60Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
61Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
62Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
63Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
Advection
Flux-based level set method (see video on my homepage)
F., Mikula: High resolution flux-based level set method; 2005
64Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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
65Kiel, 23.6.2006 peter.frolkovic@uni-hd.de
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)