On quasi-richards equation and finite volume approximation of two-phase flow with unlimited air mobility

Transcription

1 On quasi-richards equation and finite volume approximation of two-phase flow with unlimited air mobility B. Andreianov 1, R. Eymard 2, M. Ghilani 3,4 and N. Marhraoui 4 1 Université de Franche-Comte Besançon, and LJLL, Paris 6 2 Université de Marne-la-Vallée, Paris, 3 Ecole Nationale Supérieure des Arts et Metiers, Meknès (Maroc), 4 Université Moulay Ismail, Faculté des Sciences, Meknès (Maroc) February 25-27, th DFG-CNRS Workshop, LJLL, Paris

2 Outline 1 Introduction and motivation

3 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate?

4 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions?

5 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions? Robust with respect to µ finite volume approximation? Asymptotic-preserving scheme?

6 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions? Robust with respect to µ finite volume approximation? Asymptotic-preserving scheme? 2 Some theory of the quasi-richards equation

7 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions? Robust with respect to µ finite volume approximation? Asymptotic-preserving scheme? 2 Some theory of the quasi-richards equation Solutions of two-phase flow equations; estimates Passage to the limit (singular) and quasi-richards Renormalization and incomplete contraction inequality

8 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions? Robust with respect to µ finite volume approximation? Asymptotic-preserving scheme? 2 Some theory of the quasi-richards equation Solutions of two-phase flow equations; estimates Passage to the limit (singular) and quasi-richards Renormalization and incomplete contraction inequality 3 Finite volumes for two-phase flow with unlimited mobility

9 Outline 1 Introduction and motivation Two-phase model in porous medium. High/infinite mobility µ of the air phase? Is the classical Richards model appropriate? Quasi-Richards equation as the singular limit as µ Comparison with the Richards equation? Existence, uniqueness of solutions? Robust with respect to µ finite volume approximation? Asymptotic-preserving scheme? 2 Some theory of the quasi-richards equation Solutions of two-phase flow equations; estimates Passage to the limit (singular) and quasi-richards Renormalization and incomplete contraction inequality 3 Finite volumes for two-phase flow with unlimited mobility The idea of the scheme A priori estimates, existence, convergence at fixed µ Asymptotics of the scheme as µ 0 Numerical illustrations

10 Models Assumptions about groundwater flow Water and air incompressible phases Porous medium homogeneous and isotropic } Gravity neglected lower bound on saturation Source term of a special form

11 Models Assumptions about groundwater flow Water and air incompressible phases Porous medium homogeneous and isotropic } Gravity neglected lower bound on saturation Source term of a special form Richards model { ut div(k w (u) p) = s w, u = p c 1 (p atm p), Two-phase model { ut div(k w (u) p) = s w (1 u) t div(µk a (u) (p + p c (u))) = s a where µ := Ratio between the phase mobilities (we want µ )

12 Assumptions 1 Ω is a polygonal subset of R d, d = 2 or 3; T > 0 is given, 2 u m (0, 1); initial saturation u 0, source saturation c with u m u 0 (x) 1 a.e on Ω, and u m c(t, x) 1 a.e. on Ω (0, T ), 3 source s L 2, sink s L 2, s, s 0, global conservation: ( s(x, t) s(x, t))dx = 0 on (0, T ), Ω 4 k w C 0 ([0, 1]), k w non-decreasing with k w (0) = 0, k w (1) = 1 and k w (u m ) > 0, k a C 0 ([0, 1]), k a non-increasing with k a (1) = 0, k a (0) = 1 and k a (s) > 0 for all s [0, 1), p c C 0 ([u m, 1]) Lip loc ([u m, 1)), p c strictly decreasing 5 µ [1, + )

13 Mathematical setting of the problem Set f µ (u) := k w (u) k w (u) + µk a (u) ; one has f µ µ 1l [u=1] Two-phase problem: find (u, p) such that: u t div(k w (u) p) = f µ (c) s f µ (u) s on Ω (0, T ),

14 Mathematical setting of the problem Set f µ (u) := k w (u) k w (u) + µk a (u) ; one has f µ µ 1l [u=1] Two-phase problem: find (u, p) such that: u t div(k w (u) p) = f µ (c) s f µ (u) s on Ω (0, T ), (1 u) t div(µk a (u) (p + p c (u))) = (1 f µ (c)) s (1 f µ (u)) s on Ω (0, T ),

15 Mathematical setting of the problem Set f µ (u) := k w (u) k w (u) + µk a (u) ; one has f µ µ 1l [u=1] Two-phase problem: find (u, p) such that: u t div(k w (u) p) = f µ (c) s f µ (u) s on Ω (0, T ), (1 u) t div(µk a (u) (p + p c (u))) = (1 f µ (c)) s (1 f µ (u)) s on Ω (0, T ), p n = 0 on Ω (0, T ), (p + p c (u)) n = 0 on Ω (0, T ), p(x, t) dx = 0 on (0, T ) Ω

16 Mathematical setting of the problem Set f µ (u) := k w (u) k w (u) + µk a (u) ; one has f µ µ 1l [u=1] Two-phase problem: find (u, p) such that: u t div(k w (u) p) = f µ (c) s f µ (u) s on Ω (0, T ), (1 u) t div(µk a (u) (p + p c (u))) = (1 f µ (c)) s (1 f µ (u)) s on Ω (0, T ), p n = 0 on Ω (0, T ), (p + p c (u)) n = 0 on Ω (0, T ), p(x, t) dx = 0 on (0, T ) Ω u(, 0) = u 0 u m > 0 on Ω c( ) u m > 0 on Ω (0, T ).

17 Mathematical setting of the problem Set f µ (u) := k w (u) k w (u) + µk a (u) ; one has f µ µ 1l [u=1] Two-phase problem: find (u, p) such that: u t div(k w (u) p) = f µ (c) s f µ (u) s on Ω (0, T ), (1 u) t div(µk a (u) (p + p c (u))) = (1 f µ (c)) s (1 f µ (u)) s on Ω (0, T ), p n = 0 on Ω (0, T ), (p + p c (u)) n = 0 on Ω (0, T ), p(x, t) dx = 0 on (0, T ) Ω u(, 0) = u 0 u m > 0 on Ω c( ) u m > 0 on Ω (0, T ). To let µ : uniform estimates for discrete/regularized problem

18 The quasi-richards equation Theorem (Eymard, Henry, Hilhorst 09, DCDS-S 12.) There exist solutions (u µ, p µ ) for the two-phase flow problem that obey uniform estimates: lower bound u m on the saturations u µ, L 2 (0, T ; H 1 ) bound on the pressures p µ and on the 1/2-Kirchoff transform ζ(u µ ),

19 The quasi-richards equation Theorem (Eymard, Henry, Hilhorst 09, DCDS-S 12.) There exist solutions (u µ, p µ ) for the two-phase flow problem that obey uniform estimates: lower bound u m on the saturations u µ, L 2 (0, T ; H 1 ) bound on the pressures p µ and on the 1/2-Kirchoff transform ζ(u µ ), estimate on µ k a (u µ ) (p µ + p c (u µ )) 2.

20 The quasi-richards equation Theorem (Eymard, Henry, Hilhorst 09, DCDS-S 12.) There exist solutions (u µ, p µ ) for the two-phase flow problem that obey uniform estimates: lower bound u m on the saturations u µ, L 2 (0, T ; H 1 ) bound on the pressures p µ and on the 1/2-Kirchoff transform ζ(u µ ), estimate on µ k a (u µ ) (p µ + p c (u µ )) 2. Any accumulation point of (u µ, p µ ) µ as µ satisfies { ut div(k w (u) p) = s w (p + p c (u)) = 0 a.e. on the set [u < 1],

21 The quasi-richards equation Theorem (Eymard, Henry, Hilhorst 09, DCDS-S 12.) There exist solutions (u µ, p µ ) for the two-phase flow problem that obey uniform estimates: lower bound u m on the saturations u µ, L 2 (0, T ; H 1 ) bound on the pressures p µ and on the 1/2-Kirchoff transform ζ(u µ ), estimate on µ k a (u µ ) (p µ + p c (u µ )) 2. Any accumulation point of (u µ, p µ ) µ as µ satisfies { ut div(k w (u) p) = s w = s 1l [c=1] θ s 1l [u=1] (p + p c (u)) = 0 a.e. on the set [u < 1], where θ is an unknown [0, 1]-valued function.

22 The quasi-richards equation Theorem (Eymard, Henry, Hilhorst 09, DCDS-S 12.) There exist solutions (u µ, p µ ) for the two-phase flow problem that obey uniform estimates: lower bound u m on the saturations u µ, L 2 (0, T ; H 1 ) bound on the pressures p µ and on the 1/2-Kirchoff transform ζ(u µ ), estimate on µ k a (u µ ) (p µ + p c (u µ )) 2. Any accumulation point of (u µ, p µ ) µ as µ satisfies { ut div(k w (u) p) = s w = s 1l [c=1] θ s 1l [u=1] (p + p c (u)) = 0 a.e. on the set [u < 1], where θ is an unknown [0, 1]-valued function. Thus: solution of the quasi-richards eqn. is a triple (u, p, θ) with p = p c (u) on [u < 1] and with θ defined on [u = 1]. Regularity: u is [u m, 1]-valued with ζ(u) L 2 (0, T ; H 1 ), p L 2 (0, T ; H 1 ).

23 Formal justification of the estimates: Multiply first equation by p, second equation by (p + p c (u)), sum up, use chain rule on p c (u)u t, use k w (u) k w (u m ) > 0 L 2 bounds on p 2, on k a (u) µ (p + p c (u)) 2.

24 Formal justification of the estimates: Multiply first equation by p, second equation by (p + p c (u)), sum up, use chain rule on p c (u)u t, use k w (u) k w (u m ) > 0 L 2 bounds on p 2, on k a (u) µ (p + p c (u)) 2. Write a global flux formulation: div q = s s, q n = 0, u t div[ f µ (u)q k w (u) Q(u) ] = s w, Q(u) n = 0 with q = M µ (u) (p + Q(u)) and Q(u) := u 0 (1 f µ(s))p c(s) ds.

25 Formal justification of the estimates: Multiply first equation by p, second equation by (p + p c (u)), sum up, use chain rule on p c (u)u t, use k w (u) k w (u m ) > 0 L 2 bounds on p 2, on k a (u) µ (p + p c (u)) 2. Write a global flux formulation: div q = s s, q n = 0, u t div[ f µ (u)q k w (u) Q(u) ] = s w, Q(u) n = 0 with q = M µ (u) (p + Q(u)) and Q(u) := u 0 (1 f µ(s))p c(s) ds. Eliminate q from the resulting system: test fct p c (u) in the 2nd, test function F µ (u) := u 0 f µ(s)p c(s) ds in the 1st equation k w (u) Q(u) p c (u) = k a(u) µk w (u) k w (u) + µk a (u) p c(u) 2 u 2 bounded in L 1

26 Formal justification of the estimates: Multiply first equation by p, second equation by (p + p c (u)), sum up, use chain rule on p c (u)u t, use k w (u) k w (u m ) > 0 L 2 bounds on p 2, on k a (u) µ (p + p c (u)) 2. Write a global flux formulation: div q = s s, q n = 0, u t div[ f µ (u)q k w (u) Q(u) ] = s w, Q(u) n = 0 with q = M µ (u) (p + Q(u)) and Q(u) := u 0 (1 f µ(s))p c(s) ds. Eliminate q from the resulting system: test fct p c (u) in the 2nd, test function F µ (u) := u 0 f µ(s)p c(s) ds in the 1st equation k w (u) Q(u) p c (u) = k a(u) µk w (u) k w (u) + µk a (u) p c(u) 2 u 2 bounded in L 1 using k w (u) k w (u m ) > 0, we get L 1 bound on ζ(u) 2 = k w (u) p c(u) u 2 k a(u) µk w (u) k w (u) + µk a (u) p c(u) 2 u 2.

27 Justification of passage to the limit; multiplier θ In addition to estimate of ζ(u µ ), use translation estimates in time to get strong compactness of ζ(u µ ); use invertibility of ζ( ) create a strong limit u of u µ.

28 Justification of passage to the limit; multiplier θ In addition to estimate of ζ(u µ ), use translation estimates in time to get strong compactness of ζ(u µ ); use invertibility of ζ( ) create a strong limit u of u µ. Pass to strong limit in nonlinearities k a (u µ ), k w (u µ ), to weak limit in p µ create a weak limit p of p µ.

29 Justification of passage to the limit; multiplier θ In addition to estimate of ζ(u µ ), use translation estimates in time to get strong compactness of ζ(u µ ); use invertibility of ζ( ) create a strong limit u of u µ. Pass to strong limit in nonlinearities k a (u µ ), k w (u µ ), to weak limit in p µ create a weak limit p of p µ. Pass to weak- L limit in [0, 1]-valued multipliers f µ (u µ ) create a weak limit θ of f µ (u µ ). Identify the limit θ to 0 on the set [u = 1]

30 Justification of passage to the limit; multiplier θ In addition to estimate of ζ(u µ ), use translation estimates in time to get strong compactness of ζ(u µ ); use invertibility of ζ( ) create a strong limit u of u µ. Pass to strong limit in nonlinearities k a (u µ ), k w (u µ ), to weak limit in p µ create a weak limit p of p µ. Pass to weak- L limit in [0, 1]-valued multipliers f µ (u µ ) create a weak limit θ of f µ (u µ ). Identify the limit θ to 0 on the set [u = 1] Combine everything get 1st line (equation) of weak quasi-richards formulation.

31 Justification of passage to the limit; multiplier θ In addition to estimate of ζ(u µ ), use translation estimates in time to get strong compactness of ζ(u µ ); use invertibility of ζ( ) create a strong limit u of u µ. Pass to strong limit in nonlinearities k a (u µ ), k w (u µ ), to weak limit in p µ create a weak limit p of p µ. Pass to weak- L limit in [0, 1]-valued multipliers f µ (u µ ) create a weak limit θ of f µ (u µ ). Identify the limit θ to 0 on the set [u = 1] Combine everything get 1st line (equation) of weak quasi-richards formulation. On the set [u < 1] where k a (u) > 0, pass to the limit µ in L 1 bound k a (u µ )µ (p + p c (u µ )) 2 get 2nd line (constraint) of weak quasi-richards formulation.

32 Is quasi-richards well-posed? Is it different from Richards? Richards is well-posed: Alt, Luckhaus 83. L 1 contraction inequality holds. ( uniqueness, stability) Existence of sols to quasi-richards:eymard, Henry, Hilhorst. Uniqueness? Relation to the unique solution of Richards?

33 Is quasi-richards well-posed? Is it different from Richards? Richards is well-posed: Alt, Luckhaus 83. L 1 contraction inequality holds. ( uniqueness, stability) Existence of sols to quasi-richards:eymard, Henry, Hilhorst. Uniqueness? Relation to the unique solution of Richards? Theorem (A., Eymard, Ghilani, Marhraoui 12) Assume u, û are weak solutions of the quasi-richards equation corresponding to data (u 0, s) and (û 0, ŝ). Then we have the following incomplete contraction inequality : for a.e. t, t t (u û) + (t, ) (u 0 û 0 ) + + ( s ŝ) + + s. (1) Ω Ω 0 Ω 0 [u=1=û] Proof: use renormalized solutions of Plouvier-Debaigt, Gagneux.

34 Is quasi-richards well-posed? Is it different from Richards? Theorem (A., Eymard, Ghilani, Marhraoui 12 ) Assume there is no water injection: s1l [c=1] = 0 a.e. on (0, T ) Ω (with c = c(t, x) the saturation in water of the injected fluid). Then for every datum u 0 there exists a unique u such that (u, p, θ) is a solution of the quasi-richards equation.

35 Is quasi-richards well-posed? Is it different from Richards? Theorem (A., Eymard, Ghilani, Marhraoui 12 ) Assume there is no water injection: s1l [c=1] = 0 a.e. on (0, T ) Ω (with c = c(t, x) the saturation in water of the injected fluid). Then for every datum u 0 there exists a unique u such that (u, p, θ) is a solution of the quasi-richards equation. Moreover, in absence of water injection we have θ s = 0 a.e. (no water production!); and the saturation u given by quasi-richards eqn coincides with the unique solution of the Richards eqn.

36 Is quasi-richards well-posed? Is it different from Richards? Theorem (A., Eymard, Ghilani, Marhraoui 12 ) Assume there is no water injection: s1l [c=1] = 0 a.e. on (0, T ) Ω (with c = c(t, x) the saturation in water of the injected fluid). Then for every datum u 0 there exists a unique u such that (u, p, θ) is a solution of the quasi-richards equation. Moreover, in absence of water injection we have θ s = 0 a.e. (no water production!); and the saturation u given by quasi-richards eqn coincides with the unique solution of the Richards eqn. In general, we do not expect that quasi-richards and Richards coincide: Physical reasons: p atm is not the good pressure for air when air is captured by saturated water phase While uniqueness of u in the triple (u, p, θ) can be hoped for, we do not expect uniqueness of (p, θ) in the saturated set [u = 1]. More work needed to understand quasi-richards!

37 Renormalized solutions... Idea: multiply quasi-richards by a nonlinear truncation T n (u), T n 1 on [0, 1 1 n ], T n(1) = 0. Use chain rules, obtain family of evolution equations (RenEq n ) b n (u) t ϕ n (u) = s T n (u) + ψ n (u) 2 with ad hoc nonlinearities b n, ϕ n, ψ n. The information from [u < 1] is recovered as n but the information from [u = 1] is lost.

38 Renormalized solutions... Idea: multiply quasi-richards by a nonlinear truncation T n (u), T n 1 on [0, 1 1 n ], T n(1) = 0. Use chain rules, obtain family of evolution equations (RenEq n ) b n (u) t ϕ n (u) = s T n (u) + ψ n (u) 2 with ad hoc nonlinearities b n, ϕ n, ψ n. The information from [u < 1] is recovered as n but the information from [u = 1] is lost. To recover some information from [u = 1], compute t (Cstr) lim n 0 Ω ψ n(u) 2 = t 0 [u=1] ( s θ s).

39 Renormalized solutions... Idea: multiply quasi-richards by a nonlinear truncation T n (u), T n 1 on [0, 1 1 n ], T n(1) = 0. Use chain rules, obtain family of evolution equations (RenEq n ) b n (u) t ϕ n (u) = s T n (u) + ψ n (u) 2 with ad hoc nonlinearities b n, ϕ n, ψ n. The information from [u < 1] is recovered as n but the information from [u = 1] is lost. To recover some information from [u = 1], compute t (Cstr) lim n 0 Ω ψ n(u) 2 = t 0 [u=1] ( s θ s). Def. Combination of equations (RenEq n ), n and of constraint (Cstr) is called renormalized formulation of quasi-richards. Prop. A weak solution is also a renormalized solution.

40 Use of renormalized solutions. (RenEq n ) is a standard parabolic-elliptic problem L 1 -contraction ok. Given two solutions u, û, we find (b n (u) b n (û)) + L 1(t) (b(u 0 ) b(û 0 )) + L 1 t + sign + (b n (u) b n (û)) ( s T n (u) ŝ T n (û)+ ψ n (u) 2 ψ n (û) 2). 0 Ω Let n : using b n Id, we get (u û) + L 1(t) (u 0 û 0 ) + L 1 t ( ) + s 1l[u<1] ŝ 1l [û<1] + lim n 0 [u>û] t 0 u>û ( ψn (u) 2 ψ n (û) 2).

41 Use of renormalized solutions. (RenEq n ) is a standard parabolic-elliptic problem L 1 -contraction ok. Given two solutions u, û, we find (b n (u) b n (û)) + L 1(t) (b(u 0 ) b(û 0 )) + L 1 t + sign + (b n (u) b n (û)) ( s T n (u) ŝ T n (û)+ ψ n (u) 2 ψ n (û) 2). 0 Ω Let n : using b n Id, we get (u û) + L 1(t) (u 0 û 0 ) + L 1 t ( ) + s 1l[u<1] ŝ 1l [û<1] + lim n 0 [u>û] t 0 u>û Use (Cstr) trying to simplify the right-hand side......aie aie... the term t 0 [u=1=û] s survives. incomplete contraction inequality follows. ( ψn (u) 2 ψ n (û) 2).

42 Finite volume scheme Write k w (u)=f µ (u)m µ (u), M µ =k w +µk a. Set δ n+1 K,L (Z D)=Z n+1 L Z n+1 K.

43 Finite volume scheme Write k w (u)=f µ (u)m µ (u), M µ =k w +µk a. Set δ n+1 K,L (Z D)=Z n+1 The scheme is: find U D = (UK n ) n,k, P D = (PK n ) n,k satisfying U n+1 K UK n δt n m K = τ K L f µ (U n+1 K L L N K L Z n+1 K. )M µ(ūn+1 K L )δn+1 K,L (P D) + water sources

44 Finite volume scheme Write k w (u)=f µ (u)m µ (u), M µ =k w +µk a. Set δ n+1 K,L (Z D)=Z n+1 The scheme is: find U D = (UK n ) n,k, P D = (PK n ) n,k satisfying U n+1 K UK n δt n m K = τ K L f µ (U n+1 K L L N K L Z n+1 K. )M µ(ūn+1 K L )δn+1 K,L (P D) + water sources (1 U n+1 K ) (1 Un K ) δt n m K = air sources Kirchoff transform g = k a p c + τ K L (1 f µ (U n+1 K L ))M µ(ūn+1 K L )δn+1 K,L (P D) µ τ K L δ n+1 K,L (g(u D)) L N K L N K (+ discretization of IC u 0, + normalization of P D due to Neumann BC)

45 Finite volume scheme Write k w (u)=f µ (u)m µ (u), M µ =k w +µk a. Set δ n+1 K,L (Z D)=Z n+1 The scheme is: find U D = (UK n ) n,k, P D = (PK n ) n,k satisfying U n+1 K UK n δt n m K = τ K L f µ (U n+1 K L L N K L Z n+1 K. )M µ(ūn+1 K L )δn+1 K,L (P D) + water sources (1 U n+1 K ) (1 Un K ) δt n m K = air sources Kirchoff transform g = k a p c + τ K L (1 f µ (U n+1 K L ))M µ(ūn+1 K L )δn+1 K,L (P D) µ τ K L δ n+1 K,L (g(u D)) L N K L N K (+ discretization of IC u 0, + normalization of P D due to Neumann BC) where { U U n+1 n+1 K L is the upwind value : Un+1 K L = L if δ n+1 K,L (P D) 0, otherwise, U n+1 K

46 Finite volume scheme Write k w (u)=f µ (u)m µ (u), M µ =k w +µk a. Set δ n+1 K,L (Z D)=Z n+1 The scheme is: find U D = (UK n ) n,k, P D = (PK n ) n,k satisfying U n+1 K UK n δt n m K = τ K L f µ (U n+1 K L L N K L Z n+1 K. )M µ(ūn+1 K L )δn+1 K,L (P D) + water sources (1 U n+1 K ) (1 Un K ) δt n m K = air sources Kirchoff transform g = k a p c + τ K L (1 f µ (U n+1 K L ))M µ(ūn+1 K L )δn+1 K,L (P D) µ τ K L δ n+1 K,L (g(u D)) L N K L N K (+ discretization of IC u 0, + normalization of P D due to Neumann BC) where { U U n+1 n+1 K L is the upwind value : Un+1 K L = L if δ n+1 K,L (P D) 0, otherwise, Ū n+1 K L U n+1 K between Un+1 K and U n+1 L is the auxiliary value: k a (Ūn+1 K L ) δn+1 K,L (p c(u D )) = δ n+1 K,L (g(u D)) i.e., k a (Ūn+1 K L ) = g(un+1 L ) g(u n+1 K ) p c (U n+1 L ) p c (U n+1 K ).

47 Properties of the scheme the choice Ū n+1 K L makes appear µk a(ū n+1 K L )δn+1 K,L (P D p c (U D )) uniform in µ, h (discrete) estimates as for Eymard, Henry, Hilhorst... except for time translation estimate on U D (not uniform in µ)

48 Properties of the scheme the choice Ū n+1 K L makes appear µk a(ū n+1 K L )δn+1 K,L (P D p c (U D )) uniform in µ, h (discrete) estimates as for Eymard, Henry, Hilhorst... except for time translation estimate on U D (not uniform in µ) robustness even for µ 10 7 (with needed in practice) empirical convergence orders: 0.9 in natural norms (for simple tests)

49 Properties of the scheme the choice Ū n+1 K L makes appear µk a(ū n+1 K L )δn+1 K,L (P D p c (U D )) uniform in µ, h (discrete) estimates as for Eymard, Henry, Hilhorst... except for time translation estimate on U D (not uniform in µ) robustness even for µ 10 7 (with needed in practice) empirical convergence orders: 0.9 in natural norms (for simple tests) existence, compactness, convergence as h 0: as usual

50 Properties of the scheme the choice Ū n+1 K L makes appear µk a(ū n+1 K L )δn+1 K,L (P D p c (U D )) uniform in µ, h (discrete) estimates as for Eymard, Henry, Hilhorst... except for time translation estimate on U D (not uniform in µ) robustness even for µ 10 7 (with needed in practice) empirical convergence orders: 0.9 in natural norms (for simple tests) existence, compactness, convergence as h 0: as usual numerical tests: U µ D URich D 1 or const µ + residual(h) Asymptotics of the scheme as µ : a scheme for Richards?

51 Properties of the scheme the choice Ū n+1 K L makes appear µk a(ū n+1 K L )δn+1 K,L (P D p c (U D )) uniform in µ, h (discrete) estimates as for Eymard, Henry, Hilhorst... except for time translation estimate on U D (not uniform in µ) robustness even for µ 10 7 (with needed in practice) empirical convergence orders: 0.9 in natural norms (for simple tests) existence, compactness, convergence as h 0: as usual numerical tests: U µ D URich D 1 or const µ + residual(h) Asymptotics of the scheme as µ : a scheme for Richards? In the gradually saturated regime (u u M < 1) we find U n+1 K UK n δt n m K τ K L k w (U n+1 L N K L )k a(ū n+1 K k a (U n+1 K L ) K L )δn+1 K,L (P D) = 0, while the straightforward discretization of Richards equation yields k w (U n+1 K L )δn+1 K,L (P D). One can see that ka(ūn+1 1 as h 0, so we k a(u n+1 ) K L have an almost asymptotic preserving scheme: (limit µ of the two-phase scheme is a strange scheme for Richards eqn). K L )

52 Merci pour votre attention!