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COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, \'OL. IX, 267-293 (1956) Survey of the Stability Linear Finite Difference Equations* P. I>. LAX and R. D. RICHTMYEK PART I AN EQUIVALENCE THEOREM 1. Introduction Beginning with the discovery by Courant, Friedrichs and Lewy [l] of the conditional stability of certain finite difference approximations to partial differential equations, the subject of stability has been variously discussed in the literature (see bibliography at end). The present paper is concerned with the numerical solution of initial value problems by finite difference methods, generally for a finite time interval, by a sequence of calculations with increasingly finer mesh, Thus if t is the time variable and dt its increment, we are concerned with limits as At -+ 0 for fixed t, not with limits as t+co for fixed (although often the stability con- siderations are similar). The basic question is whether the solution con- verges to the true solution of the initial value problem as the mesh is refined. The term stabiZity, as usually understood, refers to a property of the finite difference equations, or rather of the above mentioned sequence of finite difference equations with increasingly finer mesh. We shall give a definition of stability in terms of the uniform boundedness a certain set of operators and then show that under suitable circumstances, for linear initial value problems, stability is necessary and sufficient for convergence in a certain uniform sense for arbitrary initial data. The circumstances are first that a certain consistency condition must be satisfied which essentially insures that the difference equations approximate the differen- tial equations under study, rather than for exampre some other differentid equations, and secondly that the initial value problem be properly posed, in a sense to be defined later. We shall not be concerned with rounding errors, and in fact assume that all arithmetic steps are carried out with infinite precision. But it will *The work for this paper was done under Contract AT-([email protected])-1480 of the Atomic Energy Commission. 267
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268 P. D. LAX, AND R. D. RICHTMYER be evident to the reader that there is an intimate connection between stability and practicality of the equations from the point of view of the growth and amplification of rounding errors. Indeed, O’Brien, Hyman and Kaplan [8] defined stability in terms of the growth of rounding errors. However, we have a slight preference for the definition given below, because it emphasizes that stability still has to be considered, even if rounding errors are negligible, unless, of course, the initial data are chosen with diabolical care so as to be exactly free of those components that would be unduly amplified if they were present. The basic notions will be spelled out ill considerable detail below in an attempt to motivate the definitions given and to justify the approach via the theory of linear operators in Banach space.
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