On stability and convergence of difference schemes for quasilinear hyperbolic initial-boundary-value problems

Abstract

In the seventies^[2,3] we obtained some results on stability of difference schemes for initial-boundary-value problems of linear diagonalized hyperbolic systems in two independent variables. Later^[4] these results were extended to general linear hyperbolic systems with "moving boundaries" and some convergence theorems were established. In [1], we completed some proofs of ‘global’ convergence of difference schemes for general quasilinear hyperbolic initial-boundary-value problems with moving boundaries. Recently, more results on convergence have been derived. From these results we know that when we solve a quasilinear hyperbolic system using certain second order Singularity-Separating difference methods^[3] (separating discontinuities, weak discontinuities etc.), the approximate solution will converge to the exact solution with a convergence rate of Δt² in L₂ norm, no matter whether or not there exist some discontinuities, such as shocks, contact discontinuities. In this paper we shall summarize our main results on this subject.

This work was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation of U.S.A.

This work was supported in part by the National Science Foundation of China.