Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations

Transcription

1 Stability of two IMEX methods, CNLF and BDF-AB, for uncoupling systems of evolution equations W. Layton a,,, C. Trenchea a,3, a Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 56, USA Abstract Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap Frog (CNLF combination and the BDF-AB combination. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. For CNLF we prove stability for the coupled system under the time step condition suggested by linear stability theory for the Leap-Frog scheme. This seems to be a first proof of a widely believed result. For BDF-AB we prove stability under a condition that is better than the one suggested by linear stability theory for the individual methods. Key words: partitioned methods, IMEX methods, CNLF, Stokes-Darcy coupling Introduction In this note we prove stability of two, second order IMEX methods for uncoupling two evolution equations with exactly skew symmetric coupling: du dt + A u + Cφ = f(t, for t > and u( = u dφ dt + A φ C T u = g(t, for t > and φ( = φ. wjl@pitt.edu, wjl. Partially supported by NSF grant DMS trenchea@pitt.edu, trenchea. Partially supported by Air Force grant FA Preprint submitted to Applied Numerical Mathematics July

2 This problem occurs, for example, after spatial discretization of the evolutionary Stokes-Darcy problem, e.g., [,5,,]. Here u : [, R N, φ : [, R M, and f, g, u, φ and the matrices A /, C have compatible dimensions (and in particular C is N M. Note especially the exactly skew symmetric coupling linking the two equations. We assume that the A i are SPD. Our analysis extends to the case of A i non-symmetric with positive definite symmetric part or even nonlinear with A(v, v Const. v. The case where A / are exactly skew symmetric, relevant to wave propagation problems with both fast and slow waves, is treated in a remark below. With superscript denoting the time step number, the first method is CNLF, the combination of Crank-Nicolson and Leap Frog given by: for n u n+ u n t φ n+ φ n t + A u n+ + u n + A φ n+ + φ n + Cφ n = f n, (CNLF C T u n = g n. Since the stability region of LF is the interval < Im(z < +, from the scalar case we expect a stability restriction of the form t λ max (C T C. Interestingly, it seems that sufficiency in the non-commutative case is not yet proven Verwer [8], Remark 3., page 6. We prove in Section that CNLF is indeed stable under (, exactly the condition suggested by the linear stability theory. For vectors of the same length, denote the usual euclidean inner product and norm by u, v := u T v, φ := φ, φ. We denote the weighted norms by u A := u T A u, and φ A := φ T A φ. Theorem (Stability of CNLF Consider CNLF. Suppose the time step restriction holds: t λ max (C T C α <, for some α <. ( Then for any n α [ u n+ + φ n+ + u n + φ n ] n ( + t u l+ + u l A l= + φ l+ + φ l A [ u + φ + u + φ ] + t [ Cφ, u Cφ, u ] n ( + t λ min(a f l + λ min(a g l. l=

3 Next we establish the stability of BDF with explicit AB coupling 3u n+ u n + u n + A u n+ + C(φ n φ n = f n+, (BDF-AB t 3φ n+ φ n + φ n + A φ n+ C T (u n u n = g n+. t The stability region of AB suggests that this combination is strictly worse than CNLF. However, we prove that the combination inherits enough stability from BDF to be stable under a time step condition that in many cases is better than the one for CNLF. Theorem (Stability of BDF-AB Consider BDF-AB. Suppose that the time step restriction holds t max{λ max (A CC T, λ max (A C T C} α <, for some α >, ( then BDF-AB is stable: u n + φ n C(initial data, forcing terms, for any n. More precisely, for all n, we have that ( u n+ + φ n+ + n ( u n+ u n + φ n+ φ n + t l= ( u + φ ( u u + φ φ ( R l+ + R l+ + t n l= + ( f l+ ( α λ min (A + gl+, λ min (A where we have denoted R l+ = tc T u l+ t (φl+ φ l + φ l + tcφ l+ + t (ul+ u l + u l, R l+ = λ/ min(a tcc T u l+ λmin(a / tcc T f l+ + λ/ min(a tc T Cφ l+ λ / min(a tc T C gl+. Note that ( assumes that A tc T C, A tcc T are SPD. 3

4 Both methods use 3 levels; approximations are needed at the first two time steps to begin. We suppose these are computed to appropriate accuracy, Verwer [8]. Because the problem and methods are linear, stability immediately implies that the error is bounded by its consistency error. Remark (The case when A i are skew symmetric. Suppose A / are skew symmetric and f = g =. The same proof shows that CNLF remains stable under exactly the same time step condition: ( α [ u n + φ n + u n + φ n ] [ u + φ + u + φ ] + t[ Cφ, u Cφ, u ]. The stability proof of BDF-AB is less clear in this case. It includes dissipation terms, which, while possibly large, do not seem to control the coupling terms. The result is a worst case stability bound with exponential growth of O(exp(βt n where β = tλ max (C T C. Our tests do show cases where the numerical dissipation dominates and drives the BDF-AB approximate solution to zero.. Previous work When A i are SPD, IMEX methods, like CNLF and BDF-AB require the solution of two, smaller SPD systems per time step (which can be done by legacy codes for the independent sub-problems as compared to one larger, nonsymmetric system for monolithically coupled methods. Given this potentially large simplification, it is not surprising that IMEX methods (and associated partitioned schemes have been used extensively in the computational practice of multi-domain, multiphysics applications. The theory of IMEX methods is also developing; see [,7,5,,] and [6] for early papers and [3,,3] and particularly [8] and the book [9] for recent work. CNLF is itself a classic (e.g. [] combination of methods in computational fluid dynamics with wide practical use, including in the dynamic core of the NCAR climate model, [5]. Partitioned methods are often motivated by available codes for subproblems [6] and tend to be application specific. Examples of partitioned methods include ones designed for fluid-structure interaction [6,7,], Maxwell s equations [9] and atmosphere-ocean coupling [,3,]. The block system we study arises in evolutionary groundwater-surface water coupling, e.g., [9,8,5,]. Mu and Zhu [] gave the first (in numerical analysis of a partitioned method based on the backward Euler-forward Euler IMEX scheme; this has been extended to, so-called, asynchronous time stepping (different time steps

5 for different system components in []. Our work herein is motivated by the search for partitioned methods for the Stokes-Darcy problem with higher accuracy and better stability. Proof of stability of CNLF This section gives a complete proof of Theorem. Lemma 3 We estimate and, if A i are SPD Cφ, u = Cφ + u u Cφ u λ / min (A u A, φ λ / min (A φ A, Cφ λ max (C T C φ. Thus Cφ, u λ max (C T C φ + λ max (C T C u. Proof. The first claim is the polarization identity. The second inequality is elementary while the fourth follows by inserting the third into the first. For the third, we have Cφ = Cφ, Cφ / = C T Cφ, φ / λ / max(c T C φ. The first of three main steps in the proof of Theorem is to take the inner product of CNLF with u n+ + u n and φ n+ + φ n and add: [ u n+ + φ n+ ] [ u n + φ n ] t t + [ ] u n+ + u n A + φ n+ + φ n A + Cφ n, u n+ + u n C T u n, φ n+ + φ n = f n, u n+ + u n + g n, u n+ + u n. (3 The second step is to rearrange the coupling terms as an exact difference between two time levels: Coupling = Cφ n, u n+ u n C T u n, φ n+ φ n = C n+/ C n /, where 5

6 C n+/ : = Cφ n, u n+ Cφ n+, u n, C n / : = Cφ n, u n Cφ n, u n. The third step is to add and subtract u n + φ n to the control the energy at level t n : [ u n+ + φ n+ + u n + φ n ] [ u n + φ n + u n + φ n ] t t + [ ] u n+ + u n A + φ n+ + φ n A + C n+/ C n / = f n, u n+ + u n + g n, u n+ + u n RHS. Using Lemma 3 we treat RHS in a standard way: RHS f n λ / min (A u n+ + u n A + g n λ / min (A φ n+ + φ n A ( λ min(a f n +λ min(a g n + ( un+ +u n A + φ n+ +φ n A. Thus, define the system energy E n+/ := Collecting terms we obtain [ u n+ + φ n+ + u n + φ n ] + tc n+/. E n+/ E n+/ + t ( u n+ + u n A + φ n+ + φ n A t(λ min(a f n + λ min(a g n. Obviously, E n+/ E n / + {positive terms} RHS immediately implies stability provided only that E n+/ > for every n. We have (using Lemma 3 to bound the coupling terms E n+/ [ u n+ + φ n+ + u n + φ n ] t λ max (C T C [ u n+ + u n + φ n+ + φ n ]. This is positive (completing the proof provided t λ max (C T C <. 3 Proof of stability of BDF-AB We proceed to prove Theorem. Take the inner product of BDF-AB with u n+, φ n+, respectively, and add. There are two keys to the proof of stability. 6

7 The first key is the treatment of the BDF term. Apply the identity [ a ] [ (a b b + ] (b c (a b + c + + = (3a b + ca with a = u n+, b = u n, c = u n, and once with a = φ n+, b = φ n, c = φ n. This gives ( u n+ + u n+ u n ( u n + u n u n ( + un+ u n +u n + ( φ n+ + φ n+ φ n ( φ n + φ n φ n + φn+ φ n + φ n + u n+ A + φ n+ A + C(φ n φ n, u n+ C T (u n u n, φ n+ = f n+, u n+ + g n+, φ n+. The second key is to rearrange the coupling terms. We use the skew-symmetry of the coupling term and the polarization identity (Lemma 3 to write it as follows: Coupling = C(φ n φ n, u n+ C T (u n u n, φ n+ (5 = C(φ n+ φ n + φ n, u n+ + C T (u n+ u n + u n, φ n+ = φn+ φ n +φ n t u n+ CC T un+ u n +u n t φ n+ C T C + Rn+. Then ( and (5 give ( u n+ + u n+ u n ( u n + u n u n + ( φ n+ + φ n+ φ n ( φ n + φ n φ n + u n+ A + φ n+ A t u n+ CC T t φn+ C T C + Rn+ = f n+, u n+ + g n+, φ n+. (6 7

8 Using again the polarization identity yields ( u n+ + u n+ u n ( u n + u n u n + ( φ n+ + φ n+ φ n ( φ n + φ n φ n + u n+ A + φ n+ A t u n+ CC T t φn+ C T C + Rn+ = λ min (A tcc T u n+ + λ min (A tcc T f n+ + λ min (A tc T C φ n+ + which by summation implies the stability result u n+ l= λ min (A tc T C gn+ R n+, + un+ u n + φn+ + φn+ φ n + u + u u + φ n ( + ( αλ min (A f n+ + + φ φ ( αλ min (A gn+ n (R l+ +R l+ l=. Numerical verification of the Theorems We give two numerical tests that confirm the theory (showing in particular that the restriction ( is sharp. The examples also illustrate that there are cases where each method s time step restriction is better than the other method. In all test cases, the initial conditions are u = (, T and φ = (, T and u, φ are computed using the implicit backward Euler. We take f = g =, so that any growth in the energy is an instability. Test. In the first case the matrices are A =, A = 3, C = yielding the following time step restrictions t CNLF =.36, t BDFAB =.99. 8

9 With the time step t =.99 t CNLF both methods are observed to be stable, Figure. With the time step t = CNLF solutions u,u,φ,φ BDF AB solutions u,u,φ,φ.5. u u u φ φ. u φ φ Fig.. Both methods stable, as predicted.. t CNLF the CNLF approximations exhibit growth and thus are unstable Since. t CNLF < t BDF AB the theory predicts BDF-AB to be stable and this is indeed seen in Figure. CNLF solutions u,u,φ,φ BDF AB solutions u,u,φ,φ.5. u u u.35 u φ φ.3 φ φ Fig.. CNLF unstable, BDF-AB stable, as predicted. Test. With matrices A =, A =, C = 5 5 the time step restrictions are t CNLF =.36, t BDFAB =.99. 9

10 With time step t =.99 t CNLF the CNLF converges, while with BDF-AB the solution is unstable, Figure 3..5 CNLF solutions u,u,φ,φ u 6 x BDF AB solutions u,u,φ,φ u u φ u φ.5 φ φ Fig. 3. CNLF stable, BDF-AB unstable, as predicted. In Figure we plot the energy at the final time (vertical axis against the time step (horizontal axis for CNLF and BDF-AB versus time-step for Test and Test, respectively. The time interval is [, ]. The x-axis length is max{ t BDF AB, t CNLF }.. ECNLF and EBDFAB energies vs. timestep 3.5 x 5 ECNLF and EBDFAB energies vs. timestep 3.5 CNLF and BDFAB energies 8 6 CNLF and BDFAB energies.5.5 tcnlf tbdfab timestep tbdfab tcnlf timestep Fig.. BDF-AB and CNLF energies: Test and Test.. Tests when A i are skew symmetric Following the remark in the introduction we test the skew symmetric case. The tests confirm the predicted stability of CNLF. The stability result for BDF-AB is less clear however. Test 3 shows a case where BDF-AB solutions grow and test shows a case when the large numerical dissipation term R n+ drives the BDF-AB solution to zero.

11 Test 3. Figure 5 plots the solutions and energy versus time step for A / = 5, t = t CNLF.99, t = t CNLF CNLF solutions u,u,φ,φ 6 x BDF AB solutions u,u,φ,φ 7 x ECNLF and EBDFAB energies vs. timestep u u u 5 u 6 φ φ φ φ CNLF and BDFAB energies tcnlf timestep Fig. 5. Test 3: CNLF, BDF-AB solutions, and BDF-AB/CNLF energies. Test. Figure 6 plots solutions and energy versus time step for A / = 5, t = t CNLF.99, t = t CNLF.. 5 The first figure in Figure 6 suggests incorrectly that CNLF experiences growth. Plotting the CNLF solution over a longer time corrects this impression; see the plot over [, 6] in Figure 7. CNLF solutions u,u,φ,φ.5 BDF AB solutions u,u,φ,φ 3 ECNLF and EBDFAB energies vs. timestep u u 3 u φ. u φ 5 φ.3 φ... CNLF and BDFAB energies tcnlf timestep Fig. 6. Test : CNLF, BDF-AB solutions, and BDF-AB/CNLF energies.

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