CIRM - Videos & books Library - A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

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Auteurs : Hu, Jingwei (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [Trans.Amer.Math.Soc. 363, no. 4 (2011): 1947-1980.] by utilizing the”spreading” property of the collision operator. In this work, we provide anew proof based on a careful L2 estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This is joint work with Kunlun Qi and Tong Yang.

Keywords : Boltzmann equation; Fourier-Galerkin spectral method; well-posedness; stability; convergence; discontinuous; filter

Codes MSC :
35Q20 - Boltzmann equations
45G10 - Other nonlinear integral equations
65M12 - Stability and convergence of numerical methods (IVP of PDE)
65M70 - Spectral, collocation and related methods

Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Jingwei_HU.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2021-03-22_Hu.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse
Organisateurs de la rencontre : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine
Dates : 22/03/2021 - 26/03/2021
Année de la rencontre : 2021
URL Congrès : https://www.chairejeanmorlet.com/2355.html

Données de citation

DOI : 10.24350/CIRM.V.19734303
Citer cette vidéo: Hu, Jingwei (2021). A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19734303
URI : http://dx.doi.org/10.24350/CIRM.V.19734303

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Bibliographie

FILBET, Francis et MOUHOT, Clément. Analysis of spectral methods for the homogeneous Boltzmann equation. Transactions of the american mathematical society, 2011, vol. 363, no 4, p. 1947-1980. - http://dx.doi.org/10.1090/S0002-9947-2010-05303-6

HU, Jingwei, QI, Kunlun, et YANG, Tong. A New Stability and Convergence Proof of the Fourier--Galerkin Spectral Method for the Spatially Homogeneous Boltzmann Equation. SIAM Journal on Numerical Analysis, 2021, vol. 59, no 2, p. 613-633. - https://doi.org/10.1137/20M1351813

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