Convergence of Finite Difference Scheme and Analytic Data

Abstract

Finite difference methods are used widely to approximate solutions of partial differential equations numerically. To build up the difference scheme from the partial differential equation, we should be careful. Formal correspondence of difference schemes to the differential equation is not sufficient for the convergence of the solutions to the exact one. For equations of hyperbolic type, the convergence is assured under the CFL condition [1], which is concerned with the ratio of the space increment to the time one. However, this does not assert that, without CFL condition, every discrete solution diverges. G. Dahlquist [2] exposed an interesting example. He considered the wave equation, and descretized it by the usual finite difference scheme. He showed that if initial data for the wave equation are real analytic, then solutions of difference scheme converge to the solution of the wave equation, even in the absence of the CFL condition. Our results in this issue is in the same direction. Our aim is to show that, in the analytic case, we need not the CFL type condition for the convergence of solutions of the finite difference scheme. We treat of the linear first order system with constant coefficients;