An implicit difference scheme for a moving boundary hyperbolic problem

Abstract

In this paper an implicit difference scheme is defined for a moving boundary hyperbolic problem, which describes a shock front propagation in a constant state. We have reformulated the problem to a fixed boundary domain where an implicit difference scheme is proposed. As is well known, the equivalent condition for the convergence of a consistent scheme is its stability. However, the only reliable methods of stability analysis are based on linear theory. Moreover, the pertinent literature provides simple examples where the linearization of a nonlinear scheme leads to incorrect stability results. On an experimental basis a discrete perturbation stability analysis was then considered. In order to investigate the convergence of the scheme we considered a particular example where an approximate similarity solution is known. In this case, we point out the numerical convergence of the scheme. Even more important is that a possible way to assess the numerical accuracy when the similarity solution does not exist is suggested.