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

Ignat LI, Pozo A, Zuazua E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

Mathematics of Computation American Mathematical Society

Book Volume: 84

Pages Range: 1633-1662

Journal Issue: 294

DOI: 10.1090/s0025-5718-2014-02915-3

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 a 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 waves. 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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

"Gheorghe Mihoc - Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy RO Romania (RO) Basque Center of Applied Mathematics (BCAM) ES Spain (ES)

How to cite

APA:

Ignat, L.I., Pozo, A., & Zuazua, E. (2015). Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws. Mathematics of Computation, 84(294), 1633-1662. https://dx.doi.org/10.1090/s0025-5718-2014-02915-3

MLA:

Ignat, Liviu I., Alejandro Pozo, and Enrique Zuazua. "Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws." Mathematics of Computation 84.294 (2015): 1633-1662.

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