Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws

TL;DR

This paper develops numerical schemes that incorporate the specific viscosity mechanisms of conservation law systems to accurately approximate physically relevant solutions, addressing limitations of standard methods.

Contribution

The authors introduce new numerical schemes that explicitly account for viscosity mechanisms, improving the physical relevance of solutions in conservation law problems.

Findings

Schemes successfully approximate physically relevant solutions.

Numerical experiments demonstrate robustness and accuracy.

Standard schemes may converge to non-physical solutions.

Abstract

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physically relevant solution. Numerical experiments illustrating the robust performance of these schemes are presented.