Analysis of spectral methods for the homogeneous Boltzmann equation

Handle URI:
http://hdl.handle.net/10754/597563
Title:
Analysis of spectral methods for the homogeneous Boltzmann equation
Authors:
Filbet, Francis; Mouhot, Clément
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). © 2010 American Mathematical Society.
Citation:
Filbet F, Mouhot C (2011) Analysis of spectral methods for the homogeneous Boltzmann equation. Transactions of the American Mathematical Society 363: 1947–1947. Available: http://dx.doi.org/10.1090/S0002-9947-2010-05303-6.
Publisher:
American Mathematical Society (AMS)
Journal:
Transactions of the American Mathematical Society
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
1-Apr-2011
DOI:
10.1090/S0002-9947-2010-05303-6
Type:
Article
ISSN:
0002-9947
Sponsors:
The first author would like to express his gratitude to the ANR JCJC-0136 MNEC (Methode Numerique pour les Equations Cinetiques) 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).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorFilbet, Francisen
dc.contributor.authorMouhot, Clémenten
dc.date.accessioned2016-02-25T12:42:07Zen
dc.date.available2016-02-25T12:42:07Zen
dc.date.issued2011-04-01en
dc.identifier.citationFilbet F, Mouhot C (2011) Analysis of spectral methods for the homogeneous Boltzmann equation. Transactions of the American Mathematical Society 363: 1947–1947. Available: http://dx.doi.org/10.1090/S0002-9947-2010-05303-6.en
dc.identifier.issn0002-9947en
dc.identifier.doi10.1090/S0002-9947-2010-05303-6en
dc.identifier.urihttp://hdl.handle.net/10754/597563en
dc.description.abstractThe 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). © 2010 American Mathematical Society.en
dc.description.sponsorshipThe first author would like to express his gratitude to the ANR JCJC-0136 MNEC (Methode Numerique pour les Equations Cinetiques) 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).en
dc.publisherAmerican Mathematical Society (AMS)en
dc.subjectAsymptotic stabilityen
dc.subjectBoltzmann equationen
dc.subjectFourier-Galerkin methoden
dc.subjectNumerical stabilityen
dc.subjectSpectral methodsen
dc.titleAnalysis of spectral methods for the homogeneous Boltzmann equationen
dc.typeArticleen
dc.identifier.journalTransactions of the American Mathematical Societyen
dc.contributor.institutionUniversite de Lyon, Lyon, Franceen
dc.contributor.institutionDepartement de Mathematiques et Applications, Paris, Franceen
kaust.grant.numberKUK-I1-007-43en
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.