Fully discrete hyperbolic initial boundary value problems with nonzero initial data

TL;DR

This paper extends stability estimates for hyperbolic initial boundary value problems to nonzero initial data, covering finite difference schemes with multiple time levels, thus broadening the applicability of existing stability theories.

Contribution

It introduces a novel approach that extends maximal stability estimates to nonzero initial data for finite difference schemes with multiple time levels.

Findings

Stability estimates valid for nonzero initial data.

Extension of Kreiss and Osher stability results.

Applicable to schemes with arbitrary time levels.

Abstract

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in \^a 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].