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.
Mots-clés : 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 },
pages = {609--633},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012042},
mrnumber = {3021700},
zbl = {1269.82058},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012042/}
}
TY - JOUR AU - Filbet, Francis AU - Rambaud, Amélie TI - Analysis of an Asymptotic Preserving Scheme for Relaxation Systems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 609 EP - 633 VL - 47 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012042/ DO - 10.1051/m2an/2012042 LA - en ID - M2AN_2013__47_2_609_0 ER -
%0 Journal Article %A Filbet, Francis %A Rambaud, Amélie %T Analysis of an Asymptotic Preserving Scheme for Relaxation Systems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 609-633 %V 47 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012042/ %R 10.1051/m2an/2012042 %G en %F 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 , Tome 47 (2013) no. 2, pp. 609-633. doi : 10.1051/m2an/2012042. http://archive.numdam.org/articles/10.1051/m2an/2012042/
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