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A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation
Virtualconference
Authors : Hu, Jingwei (Author of the conference)
CIRM (Publisher )
35Q20 - Boltzmann equations
45G10 - Other nonlinear integral equations
65M12 - Stability and convergence of numerical methods (IVP of PDE)
65M70 - Spectral, collocation and related methods
Additional resources :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Jingwei_HU.pdf
CIRM (Publisher )
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Abstract :
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
MSC Codes : Keywords : Boltzmann equation; Fourier-Galerkin spectral method; well-posedness; stability; convergence; discontinuous; filter
35Q20 - Boltzmann equations
45G10 - Other nonlinear integral equations
65M12 - Stability and convergence of numerical methods (IVP of PDE)
65M70 - Spectral, collocation and related methods
Additional resources :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Jingwei_HU.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 09/04/2021 Conference Date : 22/03/2021 Subseries : Research talks arXiv category : Analysis of PDEs ; Numerical Analysis ; Mathematical Physics Mathematical Area(s) : Numerical Analysis & Scientific Computing ; PDE Format : MP4 (.mp4) - HD Video Time : 00:42:38 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-03-22_Hu.mp4 
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Information on the Event
Event Title : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et AnalyseEvent Organizers : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine Dates : 22/03/2021 - 26/03/2021 Event Year : 2021 Event URL : https://www.chairejeanmorlet.com/2355.html
Citation Data
DOI : 10.24350/CIRM.V.19734303Cite this video as: 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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Bibliography
- 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
