On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation

TL;DR

This paper proves the stability, convergence, and long-term accuracy of equilibrium preserving spectral methods for the homogeneous Boltzmann equation, addressing previous theoretical gaps in their analysis.

Contribution

It provides the first detailed theoretical analysis of equilibrium preserving spectral methods, demonstrating their stability and spectral accuracy over time.

Findings

Proves stability and convergence of the method.

Shows spectral accuracy in long-time behavior.

Validates the method's effectiveness for steady states.

Abstract

Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.