Comptes Rendus
Partial Differential Equations/Numerical Analysis
High order asymptotic-preserving schemes for the Boltzmann equation
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 481-486.

In this Note we discuss the construction of high order asymptotic preserving numerical schemes for the Boltzmann equation. The methods are based on the use of Implicit–Explicit (IMEX) Runge–Kutta methods combined with a penalization technique recently introduced in Filbet and Jin (2010) [6].

Dans cette Note nous discutons la construction de schémas dʼordre élevé pour lʼéquation de Boltzmann qui préservent la limite asymptotique. Les méthodes sont basées sur lʼutilisation de schémas de Runge–Kutta explicites–implicites combinées avec une technique de pénalisation introduit récemment par Filbet et Jin (2010) [6].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.05.010
Giacomo Dimarco 1; Lorenzo Pareschi 2

1 Université de Toulouse, UPS, INSA, UT1, UTM, CNRS, UMR 5219, institut de mathématiques de Toulouse, 31062 Toulouse, France
2 Mathematics Department, University of Ferrara and CMCS, Ferrara, Italy
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Giacomo Dimarco; Lorenzo Pareschi. High order asymptotic-preserving schemes for the Boltzmann equation. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 481-486. doi : 10.1016/j.crma.2012.05.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.010/

[1] M. Bennoune; M. Lemou; L. Mieussens Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics, J. Comp. Phys., Volume 227 (2008), pp. 3781-3803

[2] S. Boscarino; G. Russo On a class of uniformly accurate IMEX Runge–Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comp., Volume 31 (2009), pp. 1926-1945

[3] S. Boscarino, L. Pareschi, G. Russo, Implicit–Explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comp., in press.

[4] G. Dimarco; L. Pareschi Exponential Runge–Kutta methods for stiff kinetic equations, SIAM J. Num. Anal., Volume 49 (2011), pp. 2057-2077

[5] G. Dimarco; L. Pareschi Asymptotic-preserving IMEX Runge–Kutta methods for nonlinear kinetic equations, 2012 (preprint) | arXiv

[6] F. Filbet; S. Jin A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., Volume 229 (2010), pp. 7625-7648

[7] E. Gabetta; L. Pareschi; G. Toscani Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal., Volume 34 (1997), pp. 2168-2194

[8] S. Jin Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., Volume 21 (1999), pp. 441-454

[9] M. Lemou Relaxed micro–macro schemes for kinetic equations, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 455-460

[10] C. Mouhot; L. Pareschi Fast algorithms for computing the Boltzmann collision operator, Math. Comp., Volume 75 (2006), pp. 1833-1852

[11] L. Pareschi; G. Russo Implicit–Explicit Runge–Kutta methods and applications to hyperbolic systems with relaxation, J. Sci. Comput., Volume 25 (2005), pp. 129-155

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