A NEW STABILITY AND CONVERGENCE PROOF OF THE FOURIER--GALERKIN SPECTRAL METHOD FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

6 Scopus Citations
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Detail(s)

Original languageEnglish
Pages (from-to)613–633
Journal / PublicationSIAM Journal on Numerical Analysis
Volume59
Issue number2
Online published4 Mar 2021
Publication statusPublished - 2021

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 [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217--1245] 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. Despite its practical success, the stability of the method was only recently proved in [F. Filbet and C. Mouhot, Trans. Amer. Math. Soc., 363 (2011), pp. 1947--1980] by utilizing the “spreading" property of the collision operator. In this work, we provide a new 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.

Research Area(s)

  • Boltzmann equation, Convergence, Discontinuous, Filter, Fourier-Galerkin spectral method, Stability, Well-posedness