Space–time domain decomposition methods for linear and non–linear diffusion problems Michel Kern with T.T.P . Hoang, E. Ahmed, C. Japhet, J. Roberts, J.Jaffré INRIA Paris — Maison de la Simulation Work supported by Andra & ANR Dedales EXA-DUNE — SPPEXA 2016 Symposium January 2016 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 1 / 19
Outline Motivations and problem setting 1 Linear problem 2 Non-linear problem 3 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 2 / 19
Outline Motivations and problem setting 1 Linear problem 2 Non-linear problem 3 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 3 / 19
Simulation of the transport of radionuclides around a repository Cell Access Drift Host rock Bentonite plug Backfill Vitrified waste Concrete Symetry Calculation area Far-field simulation Near-field simulation Challenges Different materials → strong heterogeneity, different time scales. Large differences in spatial scales. INRIA-SCIENTIFIQUE-UK-R Long-term computations. M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 4 / 19
Simulation of the transport of radionuclides around a repository Cell Access Drift Host rock Bentonite plug Backfill Vitrified waste Concrete Symetry Calculation area Far-field simulation Near-field simulation Challenges Different materials → strong heterogeneity, ⇒ Domain Decomposition methods different time scales. Global in Time Large differences in spatial scales. INRIA-SCIENTIFIQUE-UK-R Long-term computations. M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 4 / 19
Model problem: Simplified model for two–phase immiscible flow Fractional flow (global pressure), with Kirchoff transformation Neglect advection (focus on capillary trapping) : decouple pressure from saturation, Enchery et al. (06), Cances (08) Simplified system: Nonlinear (degenerate) diffusion equation ω∂ t S − ∆ φ ( S ) = 0 in Ω × [ 0 , T ] � S 0 λ ( u ) π ′ ( u ) du φ ( S ) = ω porosity S α water saturation λ mobility π capillary pressure (increasing) INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 5 / 19
Discontinuous capillary pressure: transmission conditions Two subdomains ¯ Ω = ¯ Ω 1 ∪ ¯ 0 . Γ = ¯ Ω 1 ∩ ¯ Ω 2 , Ω 1 ∩ Ω 2 = / Ω 2 P_c2(1) P_c1(1) Capillary pressure P_c P_c2(0) P_{c1}(0) 0 s_2 s_1 1 Saturation Transmission conditions on the interface Continuity of capillary pressure π 1 ( S 1 ) = π 2 ( S 2 ) on Γ Continuity of the flux ∇ φ 1 ( S 1 ) . n 1 = ∇ φ 2 ( S 2 ) . n 2 on Γ Chavent – Jaffré (86), Enchéry et al. (06), Cances (08), Ern et al (10), Brenner et al. (13) INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 6 / 19
Space–time domain decomposition Domain decomposition in space y x INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
Space–time domain decomposition Domain decomposition in space t y x INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
Space–time domain decomposition Domain decomposition in space t y x INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
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. INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
Space–time domain decomposition Space-time domain decomposition Domain decomposition in space t 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. INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
Space–time domain decomposition Space-time domain decomposition Domain decomposition in space t t y y x x Solve time-dependent problems in the Discretize in time and apply DD subdomains algorithm at each time step: Exchange information through the space-time interface ◮ Solve stationary problems in the subdomains Enable local discretizations both in space and in time ◮ Exchange information through the interface Minimize number of communication between subdoains Use the same time step on the whole domain. − → local time stepping INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 7 / 19
Outline Motivations and problem setting 1 Linear problem 2 Non-linear problem 3 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 8 / 19
Linear diffusion problem ◮ Time-dependent diffusion equation + homogeneous Dirichlet BC & IC c ( · , 0 ) = c 0 . ω∂ t c + div ( − D ∇ c ) = f in Ω × ( 0 , T ) , INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 9 / 19
Linear diffusion problem ◮ Time-dependent diffusion equation + homogeneous Dirichlet BC & IC c ( · , 0 ) = c 0 . ω∂ t c + div ( − D ∇ c ) = f in Ω × ( 0 , T ) , ◮ Equivalent multi-domain formulation obtained by solving subproblems ω∂ t c i + div ( − D ∇ c i ) = f in Ω i × ( 0 , T ) = 0 on ∂ Ω i ∩ ∂ Ω × ( 0 , T ) for i = 1 , 2 , c i c i ( · , 0 ) = c 0 in Ω i , with transmission conditions on space–time interface c 1 = c 2 on Γ × ( 0 , T ) . ∇ c 1 · n 1 + ∇ c 2 · n 2 = 0 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 9 / 19
Linear diffusion problem ◮ Time-dependent diffusion equation + homogeneous Dirichlet BC & IC c ( · , 0 ) = c 0 . ω∂ t c + div ( − D ∇ c ) = f in Ω × ( 0 , T ) , ◮ Equivalent multi-domain formulation obtained by solving subproblems ω∂ t c i + div ( − D ∇ c i ) = f in Ω i × ( 0 , T ) = 0 on ∂ Ω i ∩ ∂ Ω × ( 0 , T ) for i = 1 , 2 , c i c i ( · , 0 ) = c 0 in Ω i , with transmission conditions on space–time interface c 1 = c 2 on Γ × ( 0 , T ) . ∇ c 1 · n 1 + ∇ c 2 · n 2 = 0 ◮ Equivalent Robin TCs on Γ × [ 0 , T ] . For β 1 , β 2 > 0: − ∇ c 1 · n 1 + β 1 c 1 = − ∇ c 2 · n 1 + β 1 c 2 − ∇ c 2 · n 2 + β 2 c 2 = − ∇ c 1 · n 2 + β 2 c 1 INRIA-SCIENTIFIQUE-UK-R β 1 , β 2 numerical parameters, can be optimized to improve convergence rate M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 9 / 19
Schwarz waveform relation: Robin transmission conditions ◮ Robin to Robin operators, for i = 1 , 2 , j = 3 − i : S RtR : ( ξ i , f , c 0 ) → ( ∇ c i · n j + β j c i ) | Γ i where c i ( i = 1 , 2) solution of ω∂ t c i + div ( − D ∇ c i ) = f in Ω i × ( 0 , T ) − ∇ c i · n i + β i c i = ξ i on Γ × ( 0 , T ) Space – time interface problem with two Lagrange multipliers ξ 1 = S RtR ( ξ 2 , f , c 0 ) � � ξ 1 1 on Γ × [ 0 , T ] = κ R or S R ξ 2 ξ 2 = S RtR ( ξ 1 , f , c 0 ) 2 Solve with Richardson (original SWR) or GMRES Need to solve subdomain problem with Robin BC T. T. P . Hoang, J. Jaffré, C. Japhet, M. K., J.E. Roberts, Space-time domain decomposition methods for diffusion problems in mixed formulations. SIAM J. Numer. Anal., INRIA-SCIENTIFIQUE-UK-R 51(6):3532–3559, 2013. M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 10 / 19
Nonconforming discretization in time T t Information on one time grid at the interface is passed to the T other time grid at the interface ∆ t 1 using optimal L2-projections m ∆ t 2 m (Gander-Japhet-Maday-Nataf (2005)) 0 x Ω 1 Ω 2 INRIA-SCIENTIFIQUE-UK-R M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 11 / 19
Nonconforming discretization in time T t Information on one time grid at the interface is passed to the T other time grid at the interface ∆ t 1 using optimal L2-projections m ∆ t 2 m (Gander-Japhet-Maday-Nataf (2005)) 0 x Ω 1 Ω 2 Application (Andra) 2950 m 140 m 10 m 3950 m Permeability d = 510 − 12 m 2 /s in the clay layer and d = 210 − 9 m 2 /s in the repository. Non-conforming time grids: ∆ t = 2000 (years) in the repository and ∆ t = 10000 (years) INRIA-SCIENTIFIQUE-UK-R in the clay layer. M. Kern (INRIA – MdS) Space–time DD for diffusion MoMas Multiphase Days (Oct. 15) 11 / 19
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