Stable Difference Schemes with Uneven Mesh Spacings

Abstract

We consider a finite-difference approximation to the Cauchy problem for a first- order hyperbolic partial differential equation using different mesh spacings in different portions of the domain. By reformulating our problem as a difference approximation to an initial-bQundary value problem, we are able to use the theory of H. O. Kreiss and S. Osher to analyze the L2 stability of our scheme. 0. Introduction. In solving partial differential equations by finite-difference approximations, there are situations where the solution exhibits large gradients in a localized region. In such problems, one might wish to employ variable mesh patterns. The principal problem is then to obtain consistent difference equations at the interface of the mesh patterns without introducing any type of instability or without loss of overall computational efficiency. This paper considers the L2 stability of a finite-difference solution to the Cauchy problem ut = Au,, + BUV: u(x, y, 0) -(x, y), using two different spatial mesh patterns. Problemns of this type shall be referred to as mesh refinement problems in this paper. Stability is analyzed by reformulating our problem as a difference approximation to an initial-boundary value problem for a system of partial differential equations. General sufficient conditions for the stability of such systems are due to H. 0. Kreiss