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Analysis of spectral methods for the homogeneous Boltzmann equation

Authors: Francis Filbet and Clément Mouhot
Journal: Trans. Amer. Math. Soc. 363 (2011), 1947-1980
MSC (2000): Primary 82C40, 65M70, 76P05
Published electronically: November 15, 2010
MathSciNet review: 2746671
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Abstract: The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the ``spreading'' property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound).

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Additional Information

Francis Filbet
Affiliation: Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, Université de Lyon 1, INSAL, ECL, 43 boulevard 11 novembre 1918, F-69622 Villeurbanne cedex, France

Clément Mouhot
Affiliation: Centre for Mathematical Sciences, University of Cambridge, DAMTP, Wilberforce Road, Cambridge CB3 0WA, England
Address at time of publication: CNRS & École Normale Supérieure, DMA, UMR CNRS 8553, 45, rue d’Ulm, F-75230 Paris cedex 05, FRANCE

Keywords: Boltzmann equation, spectral methods, numerical stability, asymptotic stability, Fourier-Galerkin method
Received by editor(s): December 2, 2008
Published electronically: November 15, 2010
Additional Notes: The first author would like to express his gratitude to the ANR JCJC-0136 MNEC (Méthode Numérique pour les Equations Cinétiques) and to the ERC StG #239983 (NuSiKiMo) for funding.
The second author would like to thank Cambridge University who provided repeated hospitality in 2009 thanks to the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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