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/

[1] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws. Appl. Anal. 1-2 (1996) 163-193. | MR 1625520 | Zbl 0872.65086

[2] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004. | MR 1766856 | Zbl 0964.65096

[3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60 (2007) 1559-1622. | MR 2349349 | Zbl 1152.35009

[4] J.A. Carrillo, B. Yan, An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic systems for Chemotaxis. Preprint. | MR 3032835 | Zbl 1274.92007

[5] A. Chalabi, Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68 (1999) 955-970. | MR 1648367 | Zbl 0918.35088

[6] G.Q. Chen, T.P. Liu and C.D. Levermore, Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787-830. | MR 1280989 | Zbl 0806.35112

[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 2390995 | Zbl 1173.82032

[8] S. Deng, Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime. CiCp (2012). | Zbl 1265.35240

[9] G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations. To appear. SIAM J. Numer. Anal. 49 (2011) 2057-2077. | MR 2861709 | Zbl 1298.76150

[10] F. Filbet and S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources. J.Comput. Phys. 229 (2010). | MR 2674294 | Zbl 1202.82066

[11] F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model for he Boltzmann equation. J. Sci. Comput. 46 (2011). | MR 2753243 | Zbl pre05948663

[12] E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34 (1997) 2168-2194 | MR 1480374 | Zbl 0897.76071

[13] F. Golse, S. Jin and C.D. Levermore, The Convergence of Numerical Transfer Schemes in Diffusive Regimes I : The Discrete-Ordinate Method. SIAM J. Numer. Anal. 36 (1999) 1333-1369. | MR 1706766 | Zbl 1053.82030

[14] L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641-658 | MR 2004192 | Zbl 1130.82340

[15] S. Jin, L. Pareschi and G. Toscani, Diffusive Relaxation Schemes for Discrete-Velocity Kinetic Equations. SIAM J. Numer. Anal. 35 (1998) 2405-2439. | MR 1655853 | Zbl 0938.35097

[16] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. | MR 1718639 | Zbl 0947.82008

[17] A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws : convergence and error estimates. SIAM J. Math. Anal. 28 (1997) 1446-1456. | MR 1474223 | Zbl 0963.35119

[18] T.P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 1 (1987) 153-175. | MR 872145 | Zbl 0633.35049

[19] G. Naldi and L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37 (2000) 1246-1270. | MR 1756424 | Zbl 0954.35109

[20] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws. Commun. Pure Appl. Math. 8 (1996) 795-823. | MR 1391756 | Zbl 0872.35064

[21] E. Tadmor and T. Tang, Pointwise error estimates for scalar conservation laws with piecewise smooth solutions. SIAM J. Numer. Anal. 36 (1999) 1739-1758. | MR 1712177 | Zbl 0934.35088

[22] E. Tadmor and T. Tang, Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32 (2000) 870-886. | MR 1814742 | Zbl 0979.35098

[23] T. Tang and J. Wang, Convergence of MUSCL relaxing schemes to the relaxed schemes of conservation laws with stiff source terms. J. Sci. Comput. 15 (2000) 173-195. | MR 1827576 | Zbl 0982.65101