Convenient Stability Criteria for Difference Approximations of Hyperbolic Initial-Boundary Value Problems. II

The purpose of this paper is to extend the results of [4] in order to achieve more versatile, convenient stability criteria for a wide class of finite-difference approximations to initial-boundary value problems associated with the hyperbolic system u, = Aux + Bu + i in the quarter plane x > 0, / > 0. With these criteria, stability is easily established for a large number of examples, where many of the cases studied in the recent literature are included and generalized. 0. Introduction. In this paper we sharpen and extend the results of [4] in order to achieve more versatile, convenient, sufficient stability criteria for a large class of approximations to the initial-boundary value problem associated with the hyperbolic system u, = Aux + Bu + f in the quarter plane x > 0, t > 0. Our difference approximation consists of a general difference scheme—explicit or implicit, dissipative or not, two-level or multi-level—and boundary conditions of a wider type than discussed in [4]. As in [4], we restrict attention to the case where the outflow components of the principal part of the boundary conditions are translatory, i.e., determined at all boundary points by the same coefficients. Such boundary conditions are commonly used in practice; and in particular, when the boundary consists of a single point, the boundary conditions are translatory by definition. Throughout the paper we assume that the basic scheme is stable for the pure Cauchy problem, and that the other assumptions which guarantee the validity of the Gustafsson-Kreiss-Sundström stability theory in [5] hold for our case. We then raise the question of stability for our approximation in the sense of Definition 3.3 in [5]. Our stability analysis begins in Section 2, where we show (Theorem 2.1) that our entire approximation is stable if and only if the scalar outflow components of its principal part are stable. Thus, our global stability question is reduced to that of a Received March 13, 1986. 1980 Mathematics Subject Classification. Primary 65M10; Secondary 65N10. * Research sponsored in part by the Air Force Office of Scientific Research, United States Air Force Grant AFOSR-83-0150. Part of this research was carried out while the author visited the University of California in Los Angeles and Santa Barbara, and California Institute of Technology. **Research supported in part by NASA Contract NAS1-17070 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665. Additional support was provided by NSF Grant DMS85-03294 and ARO Grant DAAG29-85-K-0190 while in residence at UCLA. The author is a Bat-Sheva Foundation Fellow. ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page 503 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 504 MOSHE GOLDBERG AND EITAN TADMOR scalar, homogeneous, outflow problem which, as in [4], is the main subject of this paper. We state our stability criteria for the reduced problem in Theorems 3.1 and 3.2 of Section 3. These criteria depend both on the basic scheme and the boundary conditions, but very little on the intricate interaction between the two. It follows that our criteria provide in many cases a convenient, easily-to-check alternative to the well-known Gustafsson-Kreiss-Sundstrôm criterion in [5]. We proceed in Section 3 to use our stability criteria in Theorems 3.1 and 3.2 together with Lemmas 3.1 and 3.2 in order to establish all our previous examples in [4] as well as new ones. This includes a host of dissipative and nondissipative examples that include and generalize many of the cases studied in the recent literature; e.g., [1]-[10], [12]-[15]. As in [4], we point out that there is no difficulty in extending our stability criteria to two-boundary problems, since if the corresponding left and right quarter-plane problems are stable then, by Theorem 5.4 of [5], the original problem is stable as well. We also remark that there are no essential obstacles in extending our results to initial-boundary value problems with variable coefficients. 1. The Differential Problem and the Difference Approximation. Consider the first-order hyperbolic system of partial differential equations (1.1a) du(x,t)/at = Adu(x,t)/dx + Bu(x,t) = i(x,t), x > 0, t > 0, where u(x, t) = (um(x, t),...,u(n)(x, t))' is the unknown vector (prime denoting the transpose), i(x,t) = (f(l)(x,t),...,f{n)(x,t))' a given n-vector, and A and B fixed n X n matrices such that A is symmetric and nonsingular. Without restriction we may assume that the system is given in characteristic variables, namely A is diagonal of the form (1.2) A = [Al °A Al>0, AX1<0, \ 0 A11 J where Ax and A11 are of orders / X / and (n — I) X (n I), respectively. The solution of (1.1a) is uniquely determined if we prescribe initial values (1.1b) u(jc,0) = ù(x), x > 0, and boundary conditions (1.1c) iill(0,t) = Su1(0,t) + g(t), t>0, where S is a fixed (n I) X / coupling matrix, g(f ) a given (n I)-vector, and (1.3) ul = (um,..., «<'))', un=(«,..., I/«)' a partition of u into its outflow and inflow components, respectively, corresponding to the partition of A in (1.2). Introducing a mesh size Ax > 0, At > 0, such that A = At/Ax = constant, and using the notation v„(r) = \(vAx, t), we approximate (1.1a) by a general, consistent, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use HYPERBOLIC INITIAL-BOUNDARY VALUE PROBLEMS. II 505 two-sided, solvable basic scheme of the form s Q_xvv(t + At)= E eov„(f-oA0 + Arb„(0, v = r,r+l,..., a = 0 (1-4) Qa= E AjoE'> Emv = \v + x, a=-l,...,s, j=-r where the « X « coefficient matrices Aja are polynomials in XA and AtB, and the «-vectors b„(r) depend on i(x, t) and its derivatives. The difference equations in (1.4) have a unique solution v„(r + Ar) if we provide initial values (1.5) v„(juA0 = %(pAt), ju = 0,...,s, v = 0,1,2,..., and specify, at each time level t = pAt, ¡u = s, s + 1.boundary values v„(r + Ai), v = 0,..., r — 1. These boundary values will be determined by boundary conditions of the form (1.6a) T^\(t + At) = £ ro<"\(r o-AO + Atdv(t), a = 0 v = 0,...,r1, ^=1^, a= -l,...,<¡r, 7 = 0 where the n X n matrices C/,"' depend on A, AtB and S, and the «-vectors d„(r) are functions of f(x, t),g(t) and their derivatives. We shall assume that the leading coefficients C0(("2 X) are nonsingular, thus assuring that the boundary conditions (1.6a) can be solved for the required boundary values yy(t + At), v = r — 1,..., 0, in terms of neighboring values of v„. We shall further assume that the matrices C& depend weakly on B, in that B introduces a mere O(At) perturbation in these matrices. This assumption holds for all practical boundary conditions where the elements of Cj^ are polynomials in the entries of AtB. Finally, we assume that in accordance with the partition of A in (1.2), the C^ can be written as