Space-time domain decomposition for a nonlinear parabolic equation with discontinuous capillary pressure

Transcription

1 Space-time domain decomposition for a nonlinear parabolic equation with discontinuous capillary pressure Elyes Ahmed, Caroline Japhet, Michel Kern INRIA Paris Maison de la Simulation ENPC University Paris 13 Work supported by Andra & ANR Dedales SIAM Conference on Mathematical and Computational Issues in the Geosciences September 11 14, 2017 Erlangen, Germany M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

2 Outline 1 Motivation and model problem 2 Global in time domain decomposition 3 Discretization 4 Numerical examples M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

3 Geological disposal of nuclear waste Deep repository ( Long lived & High-level radioactive waste) Different materials strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

4 Geological disposal of nuclear waste Deep repository ( Long lived & High-level radioactive waste) Different materials strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. Use space-time (global in time) domain decomposition methods M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

5 Geological disposal of nuclear waste Deep repository ( Long lived & High-level radioactive waste) Different materials strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. Use space-time (global in time) domain decomposition methods Take into account different capillary pressure curves M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

6 Geological disposal of nuclear waste Deep repository ( Long lived & High-level radioactive waste) Different materials strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. Use space-time (global in time) domain decomposition methods Take into account different capillary pressure curves Extend optimized Schwarz method to nonlinear diffusion M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

7 Space time domain decomposition Domain decomposition in space y x M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

8 Space time domain decomposition Domain decomposition in space t y x M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

9 Space time domain decomposition Domain decomposition in space t y x M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

10 Space time domain decomposition Domain decomposition in space t y x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

11 Space time domain decomposition Domain decomposition in space t Space-time domain decomposition t y y x x Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

12 Space time domain decomposition Domain decomposition in space t Space-time domain decomposition t y y Discretize in time and apply DD algorithm at each time step: Solve stationary problems in the subdomains Exchange information through the interface Use the same time step on the whole domain. x Solve time-dependent problems in the subdomains Exchange information through the space-time interface Enable local discretizations both in space and in time local time stepping x M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

13 Model problem: nonlinear (degenerate) diffusion equation Two phase immiscible flow, global pressure + Kirchhoff transformation, neglect advection (Enchery et al. (06), Cances (08)) S: water saturation. π(s) capillary pressure, increasing function on [0, 1] (extend continuously to R). λ(s) mobility, ω porosity φ(s) = S 0 K λ(u)π (u)du. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

14 Model problem: nonlinear (degenerate) diffusion equation Two phase immiscible flow, global pressure + Kirchhoff transformation, neglect advection (Enchery et al. (06), Cances (08)) S: water saturation. π(s) capillary pressure, increasing function on [0, 1] (extend continuously to R). λ(s) mobility, ω porosity Simplified equation Two subdomains (rock types) φ(s) = S 0 K λ(u)π (u)du. ω t S φ(s) = 0 in Ω [0,T ] P_c2(1) P_c1(1) Capillary pressure P_c P_c2(0) P_{c1}(0) 0 s_2 s_1 1 Saturation M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

15 Multi-domain formulation 2 subdomains Ω 1, Ω 2, interface Γ = Ω 1 Ω 2. Solve (for i = 1,2): ω t S i φ i (S ) = 0 in Ω i [0,T ], φ i (S i ) n i = 0 on ( Ω i \Γ) (0,T ) S i (.,0) = S 0 (.) Ωi in Ω i M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

16 Multi-domain formulation 2 subdomains Ω 1, Ω 2, interface Γ = Ω 1 Ω 2. Solve (for i = 1,2): ω t S i φ i (S ) = 0 in Ω i [0,T ], φ i (S i ) n i = 0 on ( Ω i \Γ) (0,T ) S i (.,0) = S 0 (.) Ωi in Ω i together with natural transmission conditions on the space-time interface Continuity of capillary pressure π 1 (S 1 ) = π 2 (S 2 ) on Γ (0,T ) Continuity of the flux φ 1 (S 1 ).n 1 + φ 2 (S 2 ).n 2 = 0 on Γ (0,T ) M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

17 Multi-domain formulation 2 subdomains Ω 1, Ω 2, interface Γ = Ω 1 Ω 2. Solve (for i = 1,2): ω t S i φ i (S ) = 0 in Ω i [0,T ], φ i (S i ) n i = 0 on ( Ω i \Γ) (0,T ) S i (.,0) = S 0 (.) Ωi in Ω i together with natural transmission conditions on the space-time interface Continuity of capillary pressure π 1 (S 1 ) = π 2 (S 2 ) on Γ (0,T ) Continuity of the flux φ 1 (S 1 ).n 1 + φ 2 (S 2 ).n 2 = 0 on Γ (0,T ) Replace by equivalent Robin transmission conditions φ 1 (S 1 ).n 1 + β 1 π 1 (S 1 ) = φ 2 (S 2 ).n 2 + β 1 π 2 (S 2 ) on Γ (0,T ) φ 2 (S 2 ).n 2 + β 2 π 2 (S 2 ) = φ 1 (S 1 ).n 1 + β 2 π 1 (S 1 ) on Γ (0,T ) β 1,β 2 > 0 given parameters M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

18 Global in time domain decomposition algorithm Non-linear Optimized Schwarz waveform relaxation algorithm Given S 0 i, iterate for k = 0,... Solve for S k+1 i, i = 1,2,j = 3 i ω t S k+1 i φ i (S k+1 i ) = 0 in Ω i [0,T ] φ i (S k+1 i ) = 0 n i on ( Ω i \Γ) (0,T ) S k+1 i (.,0) = S 0 (.) Ωi in Ω i φ i (S k+1 i ).n i + β i π i (S k+1 i ) = φ j (S k j ).n j + β i π j (S k j ) on Γ [0,T ], β 1,β 2 can be chosen to optimize convergence rate (Bennequin-Gander-Halpern (09),Hoang-Jaffré,Japhet,Kern,Roberts (13)) Basic ingredient: subdomain solver with Robin bc. Existence: adapt proof from Enchery et al. (06), Cances (08), via convergence of finite volume scheme. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

19 Cell centered finite volume scheme (Enchery et al, 06) Triangulation T, cells K T, boundary faces σ Γ. Unknowns : cell values (S K ) K T, boundary face values (S σ ) σ EΓ Notations: K L = edge between K and L, τ K L : transmissivity Interior equation m(k ) Sn+1 K S n ( ) K + δ τ K L φ(s n+1 K ) φ(s n+1 L ) t L N (K ) + σ E Γ E K τ K,σ ( ) φ(s n+1 K ) φ(s n+1 σ ) = 0, K T. Robin BC for boundary faces ( ) τ K,σ φ(s n+1 K ) φ(s n+1 σ ) + β m(σ)π(s n+1 σ ) = g σ, σ E Γ Implemented with Matlab Reservoir Simulation Toolbox (K. A. Lie et al. (14)) Solver with automatic differentiation : no explicit programming of Jacobian M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

20 5. Nonconforming time discretizations and projections in time. We con sider semi-discrete problems in time with nonconforming time grids. Let T 1 and T 2 be two different partitions of the time interval (0, T ) into sub-intervals (see Figure 5.1). We denote by Jm i the time interval (t i m 1, t i m] and by t i m := (t i m t i m 1) fo m = 1,..., M i and i = 1, 2. We use the lowest order discontinuous Galerkin method [3, 18, 35], which is a modified backward Euler method. The same idea can be gener Use different alized to time the steps higher in order the subdomains in time case. We denote by P 0 (T i, W ) the space of piecewise Nonconforming discretization in time t T t 1 m t 2 m 0 Ω 1 Ω 2 x Figure 5.1: Nonconforming time grids in the subdomains. Information on one time grid at the interface is passed to the other time grid at the interface using L2-projections useconstant optimal functions projection in time algorithm, on grid Gander-Japhet-Maday-Nataf T i with values in W, where W (2005) = H 1 2 (Γ) for Method 1 and W = L 2 (Γ) for Method 2: P 0 (T i, W ) = { φ : (0, T ) W, φ J i m W, for m = 1,..., M i }. In order to exchange data on the space-time interface between different time grids, we define the following L 2 projection Π from P (T, W ) onto P (T, W ) (see [13, 18]) M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

21 Validation example, 2 rock types Homogenenous medium Ω = (0,1) 3. Mobility λ 0 (S) = S, S [0,1], Capillary pressure π 1 (S) = 5S 2, π 2 (S) = 5S S [0,1] Evolution of the saturation (t = 0.019,t = 0.6,t = 3) evolution of the saturation, capillary pressure, and error at final time along a line orthogonal to the interface. M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

22 DNAPL infiltration: medium with a low capillarity lens Mobilities λ o,i (S) = S 2, and λ g,i (S) = 3(1 S) 2, i {1,2}, Capilary pressure π 1 (S) = ln(1 S), and π 2 (S) = 0.5 ln(1 S). Convergence curve Evolution of saturation (t = 100,200,350,480) Influence of parameters β 1,β 2 on convergence M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12

23 Example wth 3 rock types Capillary pressure curves : π 1 (u) = 3u 2, π 2 (u) = 5u 2, and π 3 (u) = 3u Use of time windows reduces number of DD iterations (after the first one) Mesh and velocity streamlines Evolution of the saturation (t = 500, 2000, 4000) M. Kern (INRIA Paris 13) Space time DD for diffusion SIAM GS / 12