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

TL;DR

This paper presents a new stability and convergence proof for the Fourier-Galerkin spectral method applied to the spatially homogeneous Boltzmann equation, enhancing theoretical understanding of its reliability.

Contribution

The authors introduce a novel proof technique based on $L^2$ estimates, broadening the theoretical foundation for the spectral method's stability and convergence.

Findings

New stability and convergence proof established

Applicable to various initial data types

Enhanced theoretical understanding of the spectral method

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 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.