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A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

De Jingwei Hu

Apparaît dans la collection : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse

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.

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.19734303
  • Citer cette vidéo Hu, Jingwei (22/03/2021). 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
  • URL https://dx.doi.org/10.24350/CIRM.V.19734303

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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