Instability of travelling wave profiles for the Lax-Friedrichs scheme

Pages: 877 - 899, Volume 13, Issue 4, November 2005

doi:10.3934/dcds.2005.13.877       Abstract        Full Text (411.9K)       Related Articles

Paolo Baiti - Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Udine 33100, Italy (email)
Alberto Bressan - Department of Mathematics, Penn State University, University Park, Pa.16802, United States (email)
Helge Kristian Jenssen - Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States (email)

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 $n$.

Keywords:  Systems of conservation laws, discrete scheme, Lax-Friedrichs, travelling wave, instability.
Mathematics Subject Classification:  35L65, 35B05.

Received: August 2004;      Revised: February 2005;      Published: August 2005.