On the Stability of Modified Friedrichs Scheme for the Mixed Problem for Symmetric Hyperbolic System

Abstract

It is well known that the well posed Cauchy problem admits the consistent stable finite difference scheme. By the Lax's equivalence theorem the solutions of that scheme converge to the true solutions of the Cauchy problem. In the case of mixed problem these general theory is not yet established. Especially in the case of m-space variable we can find no example of consistent stable finite difference scheme. But we can say in the case of well posed mixed problem consistency and stability assures L convergence of approximate solutions to the true solution. Many authors obtained the condition for the stability of finite difference scheme with boundary condition (Strang [1], Kreiss [23, Osher [3]). But they did not find consistent stable scheme which approximate well posed mixed problem for hyperbolic system. We consider in a half space first order symmetric hyperbolic system. On the plane boundary we set dissipative boundary condition. In the interior of the region we must take the consistent scheme which is stable for the Cauchy problem. Here we consider a modified Friedrichs scheme as a most simple explicit scheme which is stable for the Cauchy problem. We can find consistent boundary scheme which assures the stability of the whole scheme.