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

Abstract

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.

Asymptotic‐preserving Godunov‐type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation terms

“…This is a very challenging test-case all the more since u is not governed by the limit equation (3) anymore. Instead, it degenerates into the solution of a diffusion equation (see [6] for instance). The results showed on figure 10 are obtained with ε = 5.10 Finally, the reference solution is obtained with a grid-converged long-time asymptoticpreserving scheme (see [7]).…”

“…During the last decade, several asymptotic-preserving schemes were proposed in the literature and we refer for instance to [10,17,13,14,9,7,6,11] for some of the most recent ones, but also to the references proposed therein. These schemes are able to give a suitable numerical approximation in the limit of both ε large and small.…”

“…The paper is focused on numerical issues, however the theory of systems of the form (1)- (2) has been carried on in [6].…”

“…Furthermore, these results are obtained for a standard CFL condition that does not depend on ε(t, x, u). The numerical method is then extended to the nonlinear cases (5)- (6) in the third section and (1)-(2) in the fourth section. Similar properties are obtained.…”

A unsplitting finite volume method for models with stiff relaxation source terms

“…Thus, a very convenient strategy to analyse this class of problems is to make use of accurate numerical methods in order to account the underlying set of solutions in equilibrium and non-equilibrium regimes. In the recent years, prominent wellbalanced and asymptotic preserving schemes were developed for solving balance laws, in particular when G(U ) = U , e.g., [3,4,6,10,11,12,13,16,21,22]). Essentially, a method is said to be well-balanced if it preserves an unperturbed steady state in a such way that properly balance the fluxes and the source at the discrete level.…”