Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

TL;DR

This paper introduces a new Fourier-Galerkin spectral method that conserves moments in kinetic equations, enhancing accuracy and stability for long-term simulations of the Boltzmann equation.

Contribution

A novel moment-preserving Fourier-Galerkin spectral method is developed, maintaining spectral accuracy and enabling efficient, conservative solutions for kinetic equations.

Findings

The method is spectrally consistent and stable.

Numerical experiments confirm theoretical properties.

The approach effectively preserves moments in simulations.

Abstract

Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.