Analysis of an Asymptotic Preserving Scheme for Relaxation Systems

Francis Filbet; Amélie Rambaud

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 2, page 609-633
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Filbet, Francis, and Rambaud, Amélie. "Analysis of an Asymptotic Preserving Scheme for Relaxation Systems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 609-633. <http://eudml.org/doc/273321>.

@article{Filbet2013,
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 (uεh, vεh) to a nonlinear relaxation system, where ε &gt; 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.},
author = {Filbet, Francis, Rambaud, Amélie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {hyperbolic equations with relaxation; fluid dynamic limit; asymptotic-preserving schemes},
language = {eng},
number = {2},
pages = {609-633},
publisher = {EDP-Sciences},
title = {Analysis of an Asymptotic Preserving Scheme for Relaxation Systems},
url = {http://eudml.org/doc/273321},
volume = {47},
year = {2013},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 2
SP - 609
EP - 633
AB - 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 &gt; 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.
LA - eng
KW - hyperbolic equations with relaxation; fluid dynamic limit; asymptotic-preserving schemes
UR - http://eudml.org/doc/273321
ER -

References

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