Analysis of spectral methods for the homogeneous Boltzmann equation

TL;DR

This paper introduces a new stability analysis method for spectral schemes solving the homogeneous Boltzmann equation, addressing the challenge of stability without positivity preservation and proving convergence under certain conditions.

Contribution

It presents a novel approach to analyze the stability of spectral methods for the Boltzmann equation using spreading properties and entropy estimates, without requiring positivity preservation.

Findings

Proves stability and convergence of spectral methods for large discretization parameters.

Develops a stability analysis framework based on collision spreading and entropy production.

Provides explicit bounds for the discretization parameter ensuring stability.

Abstract

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method is modified in order to enforce the posivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound).