Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-21T06:33:16.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of an Asymptotic Preserving Scheme for RelaxationSystems∗

Published online by Cambridge University Press:  15 January 2013

Francis Filbet  and
Amélie Rambaud
Francis Filbet
Affiliation:
Université de Lyon, UMR5208, Institut Camille Jordan, Université Claude Bernard Lyon 1 43 boulevard 11 novembre 1918, 69622 Villeurbanne Cedex, France.. filbet@math.univ-lyon1.fr; rambaud@math.univ-lyon1.fr
Amélie Rambaud
Affiliation:
Université de Lyon, UMR5208, Institut Camille Jordan, Université Claude Bernard Lyon 1 43 boulevard 11 novembre 1918, 69622 Villeurbanne Cedex, France.. filbet@math.univ-lyon1.fr; rambaud@math.univ-lyon1.fr
Get access

Abstract

We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet andS. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L.Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in thecontext of nonlinear and stiff kinetic equations. Here, we propose a convergence analysisof such a scheme for the approximation of a system of transport equations with a nonlinearsource term, for which the asymptotic limit is given by a conservation law. We investigatethe convergence of the approximate solution (uεh, vεh) to a nonlinear relaxation system, whereε > 0 is a physical parameter andh represents the discretization parameter. Uniform convergence withrespect to ε and h is proved and error estimates arealso obtained. Finally, several numerical tests are performed to illustrate the accuracyand efficiency of such a scheme.

Keywords

Hyperbolic equations with relaxationfluid dynamic limitasymptotic-preserving schemes
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 609 - 633
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Aregba-Driollet, D. and Natalini, R., Convergence of relaxation schemes for conservation laws. Appl. Anal. 1-2 (1996) 163–193. Google Scholar
Aregba-Driollet, D. and Natalini, R., Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973–2004. Google Scholar
Bianchini, S., Hanouzet, B. and Natalini, R., Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60 (2007) 1559–1622. Google Scholar
J.A. Carrillo, B. Yan, An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic systems for Chemotaxis. Preprint.
Chalabi, A., Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68 (1999) 955–970. Google Scholar
Chen, G.Q., Liu, T.P. and Levermore, C.D., Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787–830. Google Scholar
Degond, P., 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. Google Scholar
S. Deng, Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime. CiCp (2012).
Dimarco, G. and Pareschi, L., Exponential Runge-Kutta methods for stiff kinetic equations. To appear. SIAM J. Numer. Anal. 49 (2011) 2057–2077. Google Scholar
Filbet, F. and Jin, S., A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources. J.Comput. Phys. 229 (2010). Google Scholar
Filbet, F. and Jin, S., An asymptotic preserving scheme for the ES-BGK model for he Boltzmann equation. J. Sci. Comput. 46 (2011). Google Scholar
Gabetta, E., Pareschi, L. and Toscani, G., Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34 (1997) 2168–2194 Google Scholar
Golse, F., Jin, S. and Levermore, C.D., The Convergence of Numerical Transfer Schemes in Diffusive Regimes I : The Discrete-Ordinate Method. SIAM J. Numer. Anal. 36 (1999) 1333–1369. Google Scholar
Gosse, L. and Toscani, G., Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41 (2003) 641–658 Google Scholar
Jin, S., Pareschi, L. and Toscani, G., Diffusive Relaxation Schemes for Discrete-Velocity Kinetic Equations. SIAM J. Numer. Anal. 35 (1998) 2405–2439. Google Scholar
Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. Google Scholar
Kurganov, A. and Tadmor, E., Stiff systems of hyperbolic conservation laws : convergence and error estimates. SIAM J. Math. Anal. 28 (1997) 1446–1456. Google Scholar
Liu, T.P., Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 1 (1987) 153–175. Google Scholar
Naldi, G. and Pareschi, L., Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37 (2000) 1246–1270. Google Scholar
Natalini, R., Convergence to equilibrium for the relaxation approximations of conservation laws. Commun. Pure Appl. Math. 8 (1996) 795–823. Google Scholar
Tadmor, E. and Tang, T., Pointwise error estimates for scalar conservation laws with piecewise smooth solutions. SIAM J. Numer. Anal. 36 (1999) 1739–1758. Google Scholar
Tadmor, E. and Tang, T., Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32 (2000) 870–886. Google Scholar
Tang, T. and Wang, J., Convergence of MUSCL relaxing schemes to the relaxed schemes of conservation laws with stiff source terms. J. Sci. Comput. 15 (2000) 173–195. Google Scholar