Preconditioning the Newton-Krylov method for reactive transport

Transcription

1 Preconditioning the Newton-Krylov method for reactive transport Michel Kern, L. Amir, A. Taakili INRIA Paris-Rocquencourt Maison de la Simulation MoMaS workshop Reactive Transport Modeling in the Geological Sciences Institut Henri Poincaré November 2015 M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 1 / 22

2 Outline 1 Model and formulation 2 Elimination and preconditioning 3 Application to MoMaS benchmark M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 2 / 22

3 Motivations Study algorithms for reactive transport that 1 Keep chemistry and transport codes separate 2 Allow strong numerical coupling M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 3 / 22

4 Outline 1 Model and formulation 2 Elimination and preconditioning 3 Application to MoMaS benchmark M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 4 / 22

5 Model setup One phase reactive transport of N s species, undergoing N r reactions Transport operator Lc =.(uc D c) on ω R d,d = 1,2,3. u Darcy velocity, D diffusion dispersion tensor. Chemical system Aqueous and sorption reactions, all equilibrium. ( ( ) ( ) X X) Scc 0 0 S =, S cc S c c)( X X 0 M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 5 / 22

6 Model setup One phase reactive transport of N s species, undergoing N r reactions Transport operator Lc =.(uc D c) on ω R d,d = 1,2,3. u Darcy velocity, D diffusion dispersion tensor. Chemical system Aqueous and sorption reactions, all equilibrium. ( ( ) ( ) X X) Scc 0 0 S =, S cc S c c)( X X 0 Mass conservation Mass action laws φ t c +Lc = S T cc r +ST cc r, φ t c = S T c c r, ( )( ) ( ) Scc 0 logc logk =, log c log K S cc S c c M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 5 / 22

7 Elimination of equilibrium rates: coupled problem The kernel matrix (after Saaltink et al (98)) ( ) U st US T Ucc U = 0, U = c c, φ t (U cc c + U c c c) + U cc Lc = 0 0 U c c φ t U c c c = 0. Transformation T = U cc c + U c c c = T l + T s T = U c c T s Coupled system φ t T l + φ t T s + LT l = 0 φ t T = 0. Completed by mass action laws M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 6 / 22

8 Solving equilibrium chemistry System of non linear equations Mass action law Ŝ log ( ( c c) k k) = log Mass conservation Uc + Ū c = T T known from transport Take concentration logarithms as main unknowns Use globalized Newton s method (line search, trust region). M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 7 / 22

9 Solving equilibrium chemistry System of non linear equations Mass action law Ŝ log ( ( c c) k k) = log Mass conservation Uc + Ū c = T T known from transport Take concentration logarithms as main unknowns Use globalized Newton s method (line search, trust region). Role of chemical model Given total T, split into Liquid T l = Uc, Solid T s = Ū c total concentrations Result of chemical problem T s = Ψ C (T ) M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 7 / 22

10 A note on solving the equilibrium system Chemical system S logc = logk, Uc = T Using QR factorization of S T = ( ) ( ) R 1 Q 1 Q 2 gives U = Q 2. 0 Write z = logc = Q 1 z 1 + Q 2 z 2, mass action law gives R1 T z 1 = logk. Non linear system becomes Q T 2 exp(q 1z 1 + Q 2 z 2 ) = T Only unknown is z 2 (size N s N r ). Recover primary and secondary species! Jacobian is J c = Q T 2 CQ 2, with C = diag(c) = diag(exp(z)). Source of ill-conditioning is C (cf. J. Carrayrou). M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 8 / 22

11 Solution of transport by operator splitting See Chavent-Jaffré, Ackerer et al., Putti et al., Arbogast et al.,... Advection step Explicit, finite volumes Locally mass conservative Allows unstructured mesh CFL condition: use sub-time-steps Dispersion step Mixed finite elements (implicit) M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 9 / 22

12 Solution of transport by operator splitting See Chavent-Jaffré, Ackerer et al., Putti et al., Arbogast et al.,... Advection step Explicit, finite volumes Locally mass conservative Allows unstructured mesh CFL condition: use sub-time-steps Dispersion step Mixed finite elements (implicit) Condense transport solver, one time step (M + t L) C n+1 = MC n + t f n C n+1 = Ψ }{{} T (f n,c n ) A M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 9 / 22

13 Outline 1 Model and formulation 2 Elimination and preconditioning 3 Application to MoMaS benchmark M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 10 / 22

14 Solution strategies Fixed point (aka SI) Yeh-Tripathi, Carrayrou et al., Lagneau et al. + easy to program, code reuse not robust, small time steps M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 11 / 22

15 Solution strategies Fixed point (aka SI) Yeh-Tripathi, Carrayrou et al., Lagneau et al. + easy to program, code reuse not robust, small time steps Direct substitution (DSA) Lichtner et al., Saaltink et al., Steefel et al. + accurate, robust, difficult to code, large non-linear system M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 11 / 22

16 Solution strategies Fixed point (aka SI) Yeh-Tripathi, Carrayrou et al., Lagneau et al. + easy to program, code reuse not robust, small time steps Direct substitution (DSA) Lichtner et al., Saaltink et al., Steefel et al. + accurate, robust, difficult to code, large non-linear system DAE formulation Erhel and de Dieuleveult + use quality DAE software, accurate expensive M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 11 / 22

17 Solution strategies Fixed point (aka SI) Yeh-Tripathi, Carrayrou et al., Lagneau et al. + easy to program, code reuse not robust, small time steps Direct substitution (DSA) Lichtner et al., Saaltink et al., Steefel et al. + accurate, robust, difficult to code, large non-linear system DAE formulation Erhel and de Dieuleveult + use quality DAE software, accurate expensive Reduction technique Knabner et al. + Efficient, accurate, difficult to code M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 11 / 22

18 Solution strategies Fixed point (aka SI) Yeh-Tripathi, Carrayrou et al., Lagneau et al. + easy to program, code reuse not robust, small time steps Direct substitution (DSA) Lichtner et al., Saaltink et al., Steefel et al. + accurate, robust, difficult to code, large non-linear system DAE formulation Erhel and de Dieuleveult + use quality DAE software, accurate expensive Reduction technique Knabner et al. + Efficient, accurate, difficult to code (Try to) use best of SI and DS M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 11 / 22

19 A global method from the fixed-point formulation (1) Discrete non-linear system AT n+1 l + MT n+1 s T n+1 s Ψ C (T n+1 l + T n+1 s ) = 0 = M(T n l + T n s ) }{{} b n M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 12 / 22

20 A global method from the fixed-point formulation (1) Discrete non-linear system AT n+1 l + MT n+1 s T n+1 s Ψ C (T n+1 l + T n+1 s ) = 0 = M(T n l + T n s ) }{{} b n uncoupled Elimination of T l F 2 (T s ) = T s n+1 Ψ C ( Ts n+1 A 1 MT n+1 s + A 1 b n) = 0 Can be solved by block Gauss-Seidel or by Newton s method M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 12 / 22

21 A global method from the fixed-point formulation (1) Discrete non-linear system AT n+1 l + MT n+1 s T n+1 s Ψ C (T n+1 l + T n+1 s ) = 0 = M(T n l + T n s ) }{{} b n uncoupled Elimination of T l F 2 (T s ) = T s n+1 Ψ C ( Ts n+1 A 1 MT n+1 s + A 1 b n) = 0 Can be solved by block Gauss-Seidel or by Newton s method Residual computation 1 Apply Ψ T : solve transport for each species, 2 Apply Ψ C : solve chemistry for each grid cell. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 12 / 22

22 A global method from the fixed-point formulation (2) + Non-intrusive approach + Precipitation can be included One chemical equilibrium solve for each function evaluation M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 13 / 22

23 A global method from the fixed-point formulation (2) + Non-intrusive approach + Precipitation can be included One chemical equilibrium solve for each function evaluation Solution by Newton-Krylov method Solve the linear system by an iterative method (GMRES) Requires only jacobian matrix by vector products, Jacobian not stored Keep transport and chemistry as black boxes Used for CFD, shallow water, radiative transfer(keyes, Knoll, JCP 04), and for reactive transport (Hammond et al., Adv. Wat. Res. 05) Inexact Newton framework Approximation of the Newton s direction f (x k )d + f(x k ) η k f(x k ) adaptive choice of the forcing term η k (Kelley, Eisenstat and Walker) L. Amir, MK (Comp. Geosci. 09) M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 13 / 22

24 Jacobian computations Coupled formulation Jacobian with block structure ( ) A M J = D I D D is Jacobian of chemistry operator (cf. Knabner et al.: resolution function), computed implicitly Inverting Jacobian ; solve transport Elimination of T l Jacobian of F 2 : J 2 = I D + DA 1 M. Computing Jacobian: solve transport. Solve cannot use matrix form (GMRES OK) M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 14 / 22

25 Preconditioning Essential for good linear performance Difficult for matrix free formulation Block preconditioners preserve problem structure Jacobi ( ) A 0 P Jac = 0 I D ( ) P 1 Jac J = I A 1 M (I D) 1 D I Gauss-Seidel P 1 P GS = ( ) A 0 D (I D) ( ) I A GS J = 1 M 0 (I D) 1 (I D + DA 1 M) Elimination of T l equivalent to Schur comp. of Gauss-Seidel. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 15 / 22

26 Field of value analysis GMRES convergence not determined by eigenvalues (Greenbaum et al. 96) W(A) {x Ax x C n, x = 1}, convex set, contains eigenvalues of A r k 2 2 min max p(z) r 0 2. p P k z W(A) Simplified system: one species with sorption Λ(P 1 Jac J) [1 ich,1 + ich] Λ(J 2) [1,1 + Ch 2 ] Bounded away from 0 independently of h. Eingenvalues, field of values and pseudospectrum for GS preconditioning M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 16 / 22

27 Preconditioner performance 1D model (Matlab + Sundials), h = 0.05, K L = 1., σ = 1.5, and t = Mesh dependence : constant forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None Mesh dependence : adaptive forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None NI: # nonlinear iters, NLI: total # linear iters. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 17 / 22

28 Preconditioner performance 1D model (Matlab + Sundials), h = 0.05, K L = 1., σ = 1.5, and t = Mesh dependence : constant forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS Mesh dependence : adaptive forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None NI: # nonlinear iters, NLI: total # linear iters. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 17 / 22

29 Preconditioner performance 1D model (Matlab + Sundials), h = 0.05, K L = 1., σ = 1.5, and t = Mesh dependence : constant forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS Elimination Mesh dependence : adaptive forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None NI: # nonlinear iters, NLI: total # linear iters. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 17 / 22

30 Preconditioner performance 1D model (Matlab + Sundials), h = 0.05, K L = 1., σ = 1.5, and t = Mesh dependence : constant forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS Elimination Mesh dependence : adaptive forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS NI: # nonlinear iters, NLI: total # linear iters. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 17 / 22

31 Preconditioner performance 1D model (Matlab + Sundials), h = 0.05, K L = 1., σ = 1.5, and t = Mesh dependence : constant forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS Elimination Mesh dependence : adaptive forcing term h h/2 h/4 h/8 NI LI NI LI NI LI NI LI None BGS Elimination NI: # nonlinear iters, NLI: total # linear iters. M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 17 / 22

32 Outline 1 Model and formulation 2 Elimination and preconditioning 3 Application to MoMaS benchmark M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 18 / 22

33 MoMaS benchmark (easy, 1D) 9 liquid, 3 sorbed primary species. Huge variation in equilibrium constants, large stoichiometric coeffs. Long simulation time Set-up by J. Carrayrou, MK, P. Knabner (Comp. Geo. 2009) Species concentrations, t = 10 M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 19 / 22

34 Concentration profiles Left: C 1 at t = 10, middle S at t = 10, right: C 2 at t = 5010 Concentrations at right end of the domain as a function of time. Left: X3, middle: C5, right: C2 M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 20 / 22

35 Accuracy, spatial discretization n 192 n 384 n n 192 n 384 n X3 concentration at x = C5 concentration at x= Time Time Elution curves: left X 3, right C 5 Concentration of species S at t = 10, different schemes M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 21 / 22

36 Comparison of preconditioning strategies Non Linear Iterations CFT(None) CFT(JB) CFT(GS) CF(None) F Linear Iterations CFT(None) CFT(JB) CFT(GS) CF(None) F Mesh Mesh Left: Non-linear iterations, right: linear iterations M. Kern (INRIA MdS) Precond. Newton-Krylov for react. transp. MoMas Multiphase Days (Oct. 15) 22 / 22