Instability of travelling wave profiles for the Lax-Friedrichs scheme

Paolo Baiti; Alberto Bressan; Helge Kristian Jenssen

doi:10.3934/dcds.2005.13.877

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2005, Volume 13, Issue 4: 877-899. Doi: 10.3934/dcds.2005.13.877

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Instability of travelling wave profiles for the Lax-Friedrichs scheme

1.
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Udine 33100
2.
Department of Mathematics, Penn State University, University Park, Pa.16802
3.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695

Received: August 2004

Revised: February 2005

Published: July 2005

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Abstract

We study travelling wave profiles for discrete approximations to hyperbolic systems of conservation laws. A detailed example is constructed, showing that for the Lax-Friedrichs scheme the travelling profiles do not depend continuously on the wave speed, in the BV norm. Namely, taking a sequence of wave speeds $\lambda_n\to\lambda$, the corresponding profiles $\Psi_n$ converge to a limit $\Psi$ uniformly on the real line, but Tot.Var.{$\Psi_n-\Psi$}$\geq c_0>0$ for all $n$.

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Mathematics Subject Classification: 35L65, 35B05.

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