SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 613-633, January 2021.
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 $L^2$ 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.

中文翻译：

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空间齐次Boltzmann方程的Fourier-Galerkin谱方法的新稳定性和收敛性证明

SIAM数值分析学报，第59卷，第2期，第613-633页，2021年1月。
由于玻尔兹曼方程的高维，非局部和非线性碰撞积分，其数值逼近是一个具有挑战性的问题。在过去的十年中，傅里叶-加勒金光谱法[L. Pareschi和G.Russo，SIAM J. Numer。[Anal。，37（2000），pp。1217--1245]已成为解决Boltzmann方程的一种流行的确定性方法，以其高精度和快速傅里叶变换的潜力进一步证明了这一点。尽管取得了实际的成功，但该方法的稳定性直到最近才在[F. Filbet和C. Mouhot，译。阿米尔。数学。[Soc。，363（2011），pp。1947--1980]，利用了碰撞算子的“ spreading”属性，在这项工作中，我们基于对负部分的仔细$ L ^ 2 $估计提供了一个新的证明。解决方案。