Stability of Partitioned Imex Methods for Systems of Evolution Equations with Skew-symmetric Coupling

Stability is proven for an implicit-explicit, second order, two step method for uncoupling a system of two evolution equations with exactly skew symmetric coupling. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. The method proposed is an interpolation of the Crank-Nicolson Leap Frog (CNLF) combination with the BDF2-AB2 combination, being stable under the time step condition suggested by linear stability theory for the Leap-Frog scheme and BDF2-AB2. We analyze and prove the stability of a second order IMEX method for uncoupling two evolution equations with exactly skew symmetric coupling: { du dt +A1u+ Cφ = f(t), for t > 0 and u(0) = u0 dφ dt +A2φ− C u = g(t), for t > 0 and φ(0) = φ0. (1) This problem occurs, for example, after spatial discretization of the evolutionary Stokes-Darcy problem (e.g. [1, 4] and references therein). Here u : [0,∞)→ R , φ : [0,∞)→ R , and f, g, u0, φ0 and the matrices A1/2, 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 Ai are SPD. The analysis extends to the case of Ai non-symmetric with positive definite symmetric part or even nonlinear with 〈A(v), v〉 ≥ Const.|v|. The case where A1/2 are exactly skew symmetric, relevant to wave propagation problems with both fast and slow waves, is treated in Remark 2. In Theorem 1 we show that the quantities of interest for uncoupling system (1) are the following λmax(CC), min i=1,2 {λmin(Ai)}. With superscript denoting the time step number, consider the three level θ-method [2] (2θ− 2 )u n+1+(−4θ+2)un+(2θ− 2 )u n−1 ∆t +A1 ( θu + (1− θ)un−1 ) (2) + C ( 2θφ + (−2θ + 1)φn−1 ) = fn+2θ−1, (2θ− 2 )φ n+1+(−4θ+2)φn+(2θ− 3 2 )φ n−1 ∆t +A2 ( θφ + (1− θ)φn−1 ) (3) − C ( 2θu + (−2θ + 1)un−1 ) = gn+2θ−1, where θ ∈ [ 1 2 , 1 ] , and u, φ are computed with a second order method. When θ = 1 2 , one obtains the Crank-Nicolson Leap Frog (CNLF) method un+1−un−1 2∆t +A1 un+1+un−1 2 +Cφ = f, φn+1−φn−1 2∆t +A2 φn+1+φn−1 2 −Cu=g, while θ = 1 gives the BDF2-AB2 method 3 2u n+1−2un+ 2u n−1 ∆t +A1u n+1 + C ( 2φ − φn−1 ) = f, 3 2φ n+1−2φn+ 2φ n−1 ∆t +A2φ n+1 − C ( 2u − un−1 ) = g. ∗Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA, Email: [email protected]. Partially supported by Air Force grant FA 9550-09-1-0058. †This is an expanded version, containing supplementary material, of a report with the same title. 1 Remark 1. The system (2)-(3) can be rewritten as CNLF in variables u, φ, corrected with a first-order scheme (Crank-Nicolson-lagged) for the errors e = u − un−1, φ = φ − φn−1: un+1−un−1 2∆t +A1 un+1+un−1 2 +Cφ +(2θ − 1) [ en+1−en ∆t +A1 en+1+en 2 +Cφ n ] =fn+2θ−1, φn+1−φn−1 2∆t +A2 φn+1+φn−1 2 −C u+(2θ−1) [ φn+1−φn ∆t +A2 φn+1+φn 2 −C u ] =gn+2θ−1. This “modification” of CNLF introduces the numerical dissipation term (2θ − 1) ( (2θ − 1) min i=1,2 λmin(Ai)−∆tθλmax(CC) )N−2 ∑