Partial differential equations/Numerical analysis
On steady-state preserving spectral methods for homogeneous Boltzmann equations
[Sur des méthodes spectrales préservant les équilibres de l'équation de Boltzmann homogène]

Présenté par : Olivier Pironneau

Filbet, Francis1 ; Pareschi, Lorenzo2 ; Rey, Thomas3
1 Université Lyon-1 & Inria, Institut Camille-Jordan 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France
2 Mathematics and Computer Science Department, University of Ferrara, Italy
3 Center of Scientific Computation and Mathematical Modeling (CSCAMM), The University of Maryland, College Park, MD 20742-4015, USA
Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 309-314.

Dans cette note, nous présentons une construction générale de méthodes spectrales pour l'opérateur de collision de l'équation de Boltzmann permettant de préserver exactement les états stationnaires maxwelliens de ce type d'équations. Cette nouvelle approche est basée sur une décomposition de type « micro–macro » de la solution de l'équation, tout en restant très proche d'une méthode spectrale plus classique. Nous montrons que les méthodes obtenues sont capables d'approcher avec une précision spectrale, uniformément en temps, la solution de l'équation considérée, et nous présentons leur efficacité dans un test numérique.

In this note, we present a general way to construct spectral methods for the collision operator of the Boltzmann equation that preserves exactly the Maxwellian steady state of the system. We show that the resulting method is able to approximate with spectral accuracy the solution uniformly in time.

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Accepté le :
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DOI : 10.1016/j.crma.2015.01.015
Filbet, Francis 1 ; Pareschi, Lorenzo 2 ; Rey, Thomas 3

1 Université Lyon-1 & Inria, Institut Camille-Jordan 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France
2 Mathematics and Computer Science Department, University of Ferrara, Italy
3 Center of Scientific Computation and Mathematical Modeling (CSCAMM), The University of Maryland, College Park, MD 20742-4015, USA 
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Filbet, Francis; Pareschi, Lorenzo; Rey, Thomas. On steady-state preserving spectral methods for homogeneous Boltzmann equations. Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 309-314. doi : 10.1016/j.crma.2015.01.015. http://www.numdam.org/articles/10.1016/j.crma.2015.01.015/

[1] Bobylev, A.; Rjasanow, S. Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B, Fluids, Volume 18 (1999), pp. 869-887

[2] Bobylev, A. Exact solutions of the Boltzmann equation, Dokl. Akad. Nauk SSSR, Volume 225 (1975), pp. 1296-1299

[3] Cercignani, C.; Illner, R.; Pulvirenti, M. The Mathematical Theory of Dilute Gases, Springer-Verlag, 1994

[4] Dimarco, G.; Pareschi, L. Numerical methods for kinetic equations, Acta Numer., Volume 23 (2014), pp. 369-520

[5] Filbet, F.; Mouhot, C. Analysis of spectral methods for the homogeneous Boltzmann equation, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 1947-1980

[6] Filbet, F.; Mouhot, C.; Pareschi, L. Solving the Boltzmann equation in Nlog2N, SIAM J. Sci. Comput., Volume 28 (2007) no. 3, pp. 1029-1053

[7] Gamba, I.; Tharkabhushanam, S. Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., Volume 228 (2009) no. 6, pp. 2012-2036

[8] Jin, S.; Jin, S. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Mat. Univ. Parma, Volume 3 (June, 2010), pp. 177-216

[9] Jin, S.; Shi, Y. A micro–macro decomposition-based asymptotic-preserving scheme for the multispecies Boltzmann equation, SIAM J. Sci. Comput., Volume 31 (2009–2010) no. 6, pp. 4580-4606

[10] Krook, M.; Wu, T. Exact solutions of the Boltzmann equation, Phys. Fluids, Volume 20 (1977) no. 10, pp. 1589-1595

[11] Lemou, M.; Mieussens, L. A new asymptotic preserving scheme based on micro–macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., Volume 31 (2008) no. 10, pp. 334-368

[12] Mouhot, C.; Pareschi, L. Fast methods for the Boltzmann collision integral, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004) no. 1, pp. 71-76

[13] Pareschi, L.; Perthame, B. A spectral method for the homogeneous Boltzmann equation, Transp. Theory Stat. Phys., Volume 25 (1996), pp. 369-383

[14] Pareschi, L.; Russo, G. Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., Volume 37 (2000), pp. 1217-1245

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