Fast methods for the Boltzmann collision integral

Clément Mouhot; Lorenzo Pareschi

doi:10.1016/j.crma.2004.04.010

Mathematical Problems in Mechanics

Fast methods for the Boltzmann collision integral

Presented by: Carlo Cercignani

Clément Mouhot ¹; Lorenzo Pareschi ²

¹ École normale supérieure de Lyon, UMPA, 46, allée d'Italie, 69364 Lyon cedex 07, France
² Department of Mathematics, University of Ferrara, via Machiavelli 35, 35100 Ferrara, Italy

Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 71-76.

Abstract
Résumé

In this Note we present methods for the development of fast numerical schemes for the Boltzmann collision integral. These schemes are based on a combination of a Carleman-like representation together with a suitable angular approximation. For the hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. Estimates for the errors and spectral accuracy are also given.

Dans cette Note nous présentons des méthodes pour le développement de schémas numériques rapides pour l'intégrale de collision de Boltzmann. Ces schémas sont basés sur la combinaison d'une représentation proche de celle de Carleman et d'une approximation angulaire appropriée. Pour le modèle des sphères dures en dimension trois, nous en déduisons des méthodes spectrales qui peuvent être évaluées par des algorithmes rapides. Nous donnons également des estimations d'erreur et un résultat de précision spectrale.

Received: 2004-03-01
Accepted: 2004-04-27
Published online: 2004-05-28

DOI: 10.1016/j.crma.2004.04.010

Author's affiliations:

Clément Mouhot ¹; Lorenzo Pareschi ²

¹ École normale supérieure de Lyon, UMPA, 46, allée d'Italie, 69364 Lyon cedex 07, France
² Department of Mathematics, University of Ferrara, via Machiavelli 35, 35100 Ferrara, Italy

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     title = {Fast methods for the {Boltzmann} collision integral},
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     doi = {10.1016/j.crma.2004.04.010},
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Clément Mouhot; Lorenzo Pareschi. Fast methods for the Boltzmann collision integral. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 71-76. doi : 10.1016/j.crma.2004.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.010/

References
Cited by

[1] A. Bobylev; S. Rjasanow Difference scheme for the Boltzmann equation based on the fast Fourier transform, Eur. J. Mech. B Fluids, Volume 16 (1997) no. 2, pp. 293-306

[2] A.V. Bobylev; S. Rjasanow Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids, Volume 18 (1999) no. 5, pp. 869-887

[3] A.V. Bobylev; S. Rjasanow Numerical solution of the Boltzmann equation using a fully conservative difference scheme based on the fast Fourier transform, Transport Theory Statist. Phys., Volume 29 (2000) no. 3–5, pp. 289-310

[4] C. Canuto; M.Y. Hussaini; A. Quarteroni; Thomas A. Zang Spectral Methods in Fluid Dynamics, Springer Ser. Comput. Phys., Springer-Verlag, New York, 1988

[5] T. Carleman Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., Volume 60 (1932)

[6] C. Cercignani; R. Illner; M. Pulvirenti The Mathematical Theory of Dilute Gases, Appl. Math. Sci., vol. 106, Springer-Verlag, New York, 1994

[7] F. Filbet; G. Russo High order numerical methods for the space non-homogeneous Boltzmann equation, J. Comput. Phys., Volume 186 (2003) no. 2, pp. 457-480

[8] I. Ibragimov; S. Rjasanow Numerical solution of the Boltzmann equation on the uniform grid, Computing, Volume 69 (2002) no. 2, pp. 163-186

[9] C. Mouhot, L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Preprint, 2004, submitted for publication

[10] L. Pareschi Computational methods and fast algorithms for Boltzmann equations, Lecture Notes on the Discretization of the Boltzmann Equation, 2003, pp. 527-548 (Chapter 7)

[11] L. Pareschi; G. Russo; G. Toscani Fast spectral methods for the Fokker–Planck–Landau collision operator, J. Comput. Phys., Volume 165 (2000) no. 1, pp. 216-236

[12] L. Pareschi; B. Perthame A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys., Volume 25 (1996) no. 3–5, pp. 369-382

[13] L. Pareschi; G. Russo Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., Volume 37 (2000) no. 4, pp. 1217-1245

[14] L. Pareschi; G. Russo On the stability of spectral methods for the homogeneous Boltzmann equation, Transport Theory Statist. Phys., Volume 29 (2000) no. 3–5, pp. 431-447

[15] C. Villani A survey of mathematical topics in kinetic theory (S. Friedlander; D. Serre, eds.), Handbook of Fluid Mechanics, Elsevier, 2002

Cited by Sources:

^☆ Support by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282, is acknowledged.

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