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

Convenient stability criteria are obtained for difference approximations to initialboundary value problems associated with the hyperbolic system u, — Aux + Bu + t in the quarter plane x > 0, I > 0. The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outflow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well-known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized. 0. Introduction. In this paper we extend the results of [2] to obtain easily checkable stability criteria for difference approximations of initial-boundary value problems associated with the linear hyperbolic differential system u, = Aux + Bu + f in the quarter plane x > 0, t > 0. The difference approximations, introduced in Section 1, consist of arbitrary basic schemes-explicit or implicit, dissipative or unitary, two-level or multi-level-and boundary conditions of a rather general type. The first step in our stability analysis is made in Section 2, where we prove that the approximation is stable if and only if the scalar outflow components of its principal part are stable. This reduces the global stability question to that of a scalar, homogeneous, outflow problem which thereafter becomes the main object of the paper. Investigating the stability of the reduced problem, our main results are restricted to the case where 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 numerical boundary consists of a single point the boundary conditions are translatory by definition. The main stability criteria for the translatory case, stated without proof in Section 3, are given essentially in terms of the boundary conditions. Such schemeindependent criteria eliminate the need to analyze the intricate and often complicated interaction between the basic scheme and the boundary conditions; hence Received August 8, 1980. 1980 Mathematics Subject Classification. Primary 65M10; Secondary 65N10. * The research of the first author was sponsored in part by the Air Force Office of Scientific Research, Air Force System Command, United States j\ir Force Grant Nos. AFOSR-76-3046 and AFOSR-79-0127. ** The research of the second author is part of his Ph.D. dissertation and has been sponsored in part by the Air Force Office of Scientific Research (NAM) through the European Office of Aerospace Research, Air Force System Command, United States Air Force, under Grant 76-3035. © 1981 American Mathematical Society 0025-5718/81/0000-0069/$07.00 603 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 604 MOSHE GOLDBERG AND EITAN TADMOR providing convenient alternatives to the well-known stability criterion of Gustafsson, Kreiss, and Sundström [3], which is the basis for our work. As in [3], we assume that the basic scheme is stable for the pure Cauchy problem and that the approximation is solvable. Under these basic assumptions-which are obviously necessary for stability-we obtain, for example, in Theorems 3.3 and 3.4, that the reduced problem is stable if the (translatory) boundary conditions are solvable and satisfy the von Neumann condition as well as an additional simple inequality. If the basic scheme is unitary, it is also required that the boundary conditions be dissipative. Having the new stability criteria, we continue in Section 3 to study several examples. First, we reestablish the known fact that if the basic scheme is two-level and dissipative, then outflow boundary conditions determined by horizontal extrapolation always maintain stability. Surprisingly, we show that this result is false if the basic scheme is of more than two levels. Next, for arbitrary multi-level dissipative basic schemes, we find that if the outflow boundary conditions are generated, for example, by oblique extrapolation, by the Box-Scheme, or by the right-sided Euler scheme, then overall stability is assured. Finally, for basic schemes (dissipative or unitary), we show that overall stability holds if the outflow boundary conditions are determined by the right-sided explicit or implicit Euler schemes. These examples incorporate many special cases discussed in recent literature [l]-[4], [6], [9], [10]. In Sections 4 and 5 we prove the results stated in Section 3. It should be pointed out that there is no difficulty in extending our stability criteria to cases with two boundaries. In fact, if the corresponding left and right quarter-plane problems are stable, then, by Theorem 5.4 of [3], the original two-boundary problem is stable as well. Thanks are due to Björn Engquist and Stanley Osher for most helpful discussions. 1. The Difference Approximation. Consider the first order hyperbolic system of partial differential equations (1.1a) du(x, t)/at = Adu(x, t)/ax + Bu(x, t) + t(x, t), x > 0, t > 0, where u(x, t) = (uw(x, t), . . . , u(n\x, /))' is the vector of unknowns (prime denoting the transpose), t(x, t) = (/(I)(jc, 0» • • • >/(n)(*> 0)' is a given vector, and A and B are fixed n X n matrices so that A is Hermitian and nonsingular. Without restriction we may assume that A is diagonal of the form (1.2) A-ft1 j¡„), ̂ <0,„»»>0, where A ' ' and A " " are of orders / X / and (n I) X (n — I), respectively. The solution of (1.1) is uniquely determined if we prescribe initial values (Lib) u(x, 0) = û(jc), x > 0, and boundary conditions (Lie) u'(0, t) = Su"(0, /) + g(t), t > 0, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use STABILITY OF DIFFERENCE APPROXIMATIONS 605 where S is a fixed / X (n I) matrix, g(t) is a given /-vector, and (1.3) u1 = («<»>, . . . , «<'>)', u" = (u«+x\ ..., !!<»>)' is a partition of u into inflow and outflow unknowns, respectively, corresponding to the partition of A. In order to solve the initial-boundary value problem (1.1) by difference approximations we introduce a mesh size h = Ax > 0, k = At > 0, such that X = h/k = constant. Using the notation v„(i) = \(vh, t), we approximate (1.1a) by a consistent, two-sided, general multi-step basic scheme of the form o-iV„0 + k) = 2 &▼,(' ok) + kby(t), v = r,r+l,..., (1.4a) °-° p Q„ = 2 AJaEj, £v„ = v„+1, a = -1, , s, j--r where the n X n matrices Aja are polynomials in A and .to, and the «-vectors b„(f) depend smoothly on fix, t) and its derivatives. To solve (L4a) uniquely, we provide initial values (1.4b) yv(ok) = vy(ok), o = 0, ...,s,p = 0,1, ..., where in addition we must specify, at each time step t = ok > sk, boundary values \p(t + k), ¡i = 0, . . . , r — 1. The required boundary values will be determined by two sets of boundary conditions, the first of which is obtained by taking the last n — I components of general boundary conditions of the form T<_%(t + k)=2 Ti"\(t ok) + kd^t), o = 0 m TTM = 2 cjpEf M 0,.... r 1, a -1,.... q, j-o where the matrices Cj-J^ are polynomials in A and kB, the Ci!?« are nonsingular, and the «-vectors d^f) are functions of i(x, t), g(t) and their derivatives. If we put CL"> = rii(M) rni((x) r II 100 r II III» *-" in *-in \ = dM = 111