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STABILITY OF HYPERBOLIC FINITE-DIFFERENCE MODELS WITH ONE OR TWO BOUNDARIES

IP.com Disclosure Number: IPCOM000128244D
Original Publication Date: 1983-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 12 page(s) / 38K

Publishing Venue

Software Patent Institute

Related People

Lloyd N. Trefethen: AUTHOR [+3]

Abstract

The stability of finite difference models of hyperbolic partial differential equations depends on how numerical waves propagate and reflect at boundaries. This paper presents an extended numerical example illustrating the key points of this theory. s w

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

STABILITY OF HYPERBOLIC FINITE-DIFFERENCE MODELS WITH ONE OR TWO BOUNDARIES

BY *Lloyd N. Trefethen

November 1983 Report #" 9':7 [To appear in S. Osher (ed.), Proceedings of the 1983 AMS- SIAM Summer Session on Large-scale Computations in Fluid Mechanics, AMS Lectures in Applied Mathematics series.] AMS(MOS)subject classif ication:66M10 *Supported by an NSF Mathematical Sciences Postdoctoral Fellowship and by the U.S. Department of Energy under Contract DE-AC02-76-ER03077-V

Abstract:

The stability of finite difference models of hyperbolic partial differential equations depends on how numerical waves propagate and reflect at boundaries. This paper presents an extended numerical example illustrating the key points of this theory. s

w

0. introduction

In the numerical solution of hyperbolic partial differential equations by finite differ- ences, stability is well known to be a critical issue. As a first step the difference model must satisfy the von Neumann condition - that is, the basic formula should admit no exponentially growing Fourier modes. For linear problems with smoothly varying coeffi. s dents and no boundaries, this is essentially the whole story, and in fact if one rules out algebraically as well as exponentially growing local Fourier modes, then stability is assured. Results of this kind are widely known and are discussed in the superb book by Richtrnyer and Morton j9J. When boundaries are introduced, the stability problem becomes more subtle. Even here the literature is copious, and a dozen or more people have made substantial contribu- tions, including Godunov and Ryabenkii, Strung, Osher j7J, Kreiss, Gustafsson, Sundstr8m, Tadmor, and Michelson. The best known paper in this area is the one by Gustafsson, Kreiss, and Sundstr8m in 1972 [4J, which presents what is now often referred to as the "GKS stability theory". The great strength of the GKS paper is that it estab- lishes a necessary and sufficient stability condition for difference models of very general form - three-point or multipoint stencil in space, two- level or multilevel in time, explicit or implicit, dissipative or nondissipative. A difficulty with the paper is that it is very hard to read, and this has regrettably limited its influence. Fortunately, some more accessible accounts have appeared recently, including the report by Gustafsson in this volume. My own work in this field has been concerned with giving the stability question for initial boundary value problem models a physical interpretation based on the ideas of dispersive wave propagation and group velocity: Group velocity effects in finite- difference modeling have been surveyed by me in j9J and by Vichnevetsky and Bowles in [13]; others who have been interested in these matters include Matsuno, Grotjahn, and O'Brien in meteorology; Alfold, Bamberger, and Martineau-Niooletis in geophysics; Kentzer, Giles, and Thompkins in aerodynamics...