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Analysis of spectral methods for the homogeneous Boltzmann equation

Filbet, Francis; Mouhot, Clément (2011), Analysis of spectral methods for the homogeneous Boltzmann equation, Transactions of the American Mathematical Society, 363, 4, p. 1947-1980. http://dx.doi.org/10.1090/S0002-9947-2010-05303-6

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00339426/en/
Date
2011
Journal name
Transactions of the American Mathematical Society
Volume
363
Number
4
Publisher
American Mathematical Society
Pages
1947-1980
Publication identifier
http://dx.doi.org/10.1090/S0002-9947-2010-05303-6
Metadata
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Author(s)
Filbet, Francis
Mouhot, Clément
Abstract (EN)
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 is modified in order to enforce the posivity 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).
Subjects / Keywords
Boltzmann equation; spectral methods; numerical stability; asymptotic stability; Fourier-Galerkin method

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