A framework for late-time/stiff relaxation asymptotics

TL;DR

This paper develops a framework for analyzing late-time behavior of solutions to nonlinear hyperbolic systems with stiff relaxation, deriving a parabolic effective system, and introduces a finite volume scheme that maintains asymptotic accuracy under standard CFL conditions.

Contribution

The authors propose a new asymptotic framework for stiff relaxation systems and design a finite volume scheme that preserves late-time behavior with standard stability conditions.

Findings

Derivation of a parabolic-type effective system for late-time asymptotics.

Introduction of a finite volume scheme that preserves asymptotic regimes.

Scheme stability under classical CFL condition without additional restrictions.

Abstract

We consider solutions to nonlinear hyperbolic systems of balance laws with stiff relaxation and formally derive a parabolic-type effective system describing the late-time asymptotics of these solutions. We show that many examples from continuous physics fall into our framework, including the Euler equations with (possibly nonlinear) friction. We then turn our attention to the discretization of these stiff problems and introduce a new finite volume scheme which preserves the late-time asymptotic regime. Importantly, our scheme requires only the classical CFL (Courant, Friedrichs, Lewy) condition associated with the hyperbolic system under consideration, rather than the more restrictive, parabolic-type stability condition.