The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves

TL;DR

This paper proves the well-posedness of the null-timelike boundary problem for quasilinear scalar waves using energy estimates, offering insights for stable numerical algorithms and potential applications to other problems.

Contribution

It establishes the well-posedness of the null-timelike boundary problem for quasilinear waves in characteristic coordinates, introducing a new technique with broader applicability.

Findings

Proves well-posedness of the problem using energy estimates.

Provides a framework for developing stable finite difference algorithms.

Introduces a new technique applicable to other characteristic problems.

Abstract

The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.