Analysis of an Asymptotic Preserving Scheme for Relaxation Systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, p. 609-633

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 (uεh, vεh) 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.

DOI : https://doi.org/10.1051/m2an/2012042
Classification:  35L02,  82C70,  65M06
Keywords: hyperbolic equations with relaxation, fluid dynamic limit, asymptotic-preserving schemes
@article{M2AN_2013__47_2_609_0,
     author = {Filbet, Francis and Rambaud, Am\'elie},
     title = {Analysis of an Asymptotic Preserving Scheme for Relaxation Systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     pages = {609-633},
     doi = {10.1051/m2an/2012042},
     zbl = {1269.82058},
     mrnumber = {3021700},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_2_609_0}
}
Filbet, Francis; Rambaud, Amélie. Analysis of an Asymptotic Preserving Scheme for Relaxation Systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 609-633. doi : 10.1051/m2an/2012042. http://www.numdam.org/item/M2AN_2013__47_2_609_0/

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