Analysis of an Asymptotic Preserving Scheme for Relaxation Systems^∗

Université de Lyon, UMR5208, Institut Camille Jordan, Université Claude Bernard Lyon 1 43 boulevard 11 novembre 1918, 69622 Villeurbanne Cedex, France.
filbet@math.univ-lyon1.fr; rambaud@math.univ-lyon1.fr

Received: 5 January 2012
Revised: 7 August 2012

Abstract

We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet and S. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L. Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in the context of nonlinear and stiff kinetic equations. Here, we propose a convergence analysis of such a scheme for the approximation of a system of transport equations with a nonlinear source term, for which the asymptotic limit is given by a conservation law. We investigate the convergence of the approximate solution (u^ε_h, v^ε_h) to a nonlinear relaxation system, where ε > 0 is a physical parameter and h represents the discretization parameter. Uniform convergence with respect to ε and h is proved and error estimates are also obtained. Finally, several numerical tests are performed to illustrate the accuracy and efficiency of such a scheme.