Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws

TL;DR

This paper investigates the long-term behavior of numerical schemes for scalar conservation laws, showing how different schemes lead to either self-similar N-waves or viscous profiles, depending on numerical viscosity.

Contribution

It demonstrates the influence of numerical viscosity on the asymptotic behavior of schemes for scalar conservation laws, especially for the Burgers equation.

Findings

Engquist-Osher and Godunov schemes produce N-wave asymptotics.

Lax-Friedrichs scheme results in viscous self-similar profiles.

Numerical experiments confirm theoretical predictions.

Abstract

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented.