Asymptotic-preserving schemes for the initial-boundary value problem of hyperbolic relaxation systems

TL;DR

This paper introduces an asymptotic-preserving numerical scheme for hyperbolic relaxation systems with boundary conditions, effectively capturing equilibrium states and boundary layers on coarse meshes, extending previous methods to initial-boundary value problems.

Contribution

It extends asymptotic-preserving schemes to IBVPs for hyperbolic systems, enabling accurate solutions with coarse meshes and boundary layers.

Findings

The scheme accurately captures equilibrium behavior.

It effectively handles boundary layers.

The method is applicable to interface problems.

Abstract

In this work, we present a numerical method for the initial-boundary value problem (IBVP) of first-order hyperbolic systems with source terms. The scheme directly solves the relaxation system using a relatively coarse mesh and captures the equilibrium behavior quite well, even in the presence of boundary layers. This method extends the concept of asymptotic-preserving schemes from initial-value problems to IBVPs. Moreover, we apply this idea to design a unified numerical scheme for the interface problem of relaxation systems.