Late-time/stiff relaxation asymptotic-preserving approximations of hyperbolic equations

TL;DR

This paper develops asymptotic-preserving numerical methods for nonlinear hyperbolic systems with stiff relaxation, capturing late-time behavior and deriving effective equations through asymptotic expansions and discretizations.

Contribution

It introduces a Chapman-Enskog-type expansion for hyperbolic systems with stiff relaxation and proposes a finite volume discretization that preserves the late-time asymptotic limit.

Findings

The effective system accurately describes late-time behavior.

The discretization recovers the asymptotic system numerically.

Numerical experiments validate the theoretical approach.

Abstract

We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an effective system of equations describing the late-time/stiff relaxation singular limit. The structure of this new system is discussed and the role of a mathematical entropy is emphasized. Second, we propose a new finite volume discretization which, in late-time asymptotics, allows us to recover a discrete version of the same effective asymptotic system. This is achieved provided we suitably discretize the relaxation term in a way that depends on a matrix-valued free-parameter, chosen so that the desired asymptotic behavior is obtained. Our results are illustrated with several models of interest in continuum physics, and numerical experiments demonstrate the relevance of the proposed theory and numerical strategy.