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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2012042/
[1] and , Convergence of relaxation schemes for conservation laws. Appl. Anal. 1-2 (1996) 163-193. | MR | Zbl
[2] and , Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004. | MR | Zbl
[3] , and , Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60 (2007) 1559-1622. | MR | Zbl
[4] , , An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic systems for Chemotaxis. Preprint. | MR | Zbl
[5] , Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68 (1999) 955-970. | MR | Zbl
[6] , and , Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787-830. | MR | Zbl
[7] P. Degond, J.-G. Liu and M-H Vignal, Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit. SIAM J. Numer. Anal. 46 (2008) 1298-1322. | MR | Zbl
[8] , Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime. CiCp (2012). | Zbl
[9] and , Exponential Runge-Kutta methods for stiff kinetic equations. To appear. SIAM J. Numer. Anal. 49 (2011) 2057-2077. | MR | Zbl
[10] and , A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources. J.Comput. Phys. 229 (2010). | MR | Zbl
[11] and , An asymptotic preserving scheme for the ES-BGK model for he Boltzmann equation. J. Sci. Comput. 46 (2011). | MR
[12] , and , Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34 (1997) 2168-2194 | MR | Zbl
[13] , and , The Convergence of Numerical Transfer Schemes in Diffusive Regimes I : The Discrete-Ordinate Method. SIAM J. Numer. Anal. 36 (1999) 1333-1369. | MR | Zbl
[14] and , Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641-658 | MR | Zbl
[15] , and , Diffusive Relaxation Schemes for Discrete-Velocity Kinetic Equations. SIAM J. Numer. Anal. 35 (1998) 2405-2439. | MR | Zbl
[16] , Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. | MR | Zbl
[17] and , Stiff systems of hyperbolic conservation laws : convergence and error estimates. SIAM J. Math. Anal. 28 (1997) 1446-1456. | MR | Zbl
[18] , Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 1 (1987) 153-175. | MR | Zbl
[19] and , Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37 (2000) 1246-1270. | MR | Zbl
[20] , Convergence to equilibrium for the relaxation approximations of conservation laws. Commun. Pure Appl. Math. 8 (1996) 795-823. | MR | Zbl
[21] and , Pointwise error estimates for scalar conservation laws with piecewise smooth solutions. SIAM J. Numer. Anal. 36 (1999) 1739-1758. | MR | Zbl
[22] and , Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32 (2000) 870-886. | MR | Zbl
[23] and , Convergence of MUSCL relaxing schemes to the relaxed schemes of conservation laws with stiff source terms. J. Sci. Comput. 15 (2000) 173-195. | MR | Zbl
Cité par Sources :
