A Fourier spectral method for the cutoff Boltzmann equation: Convergence analysis and numerical simulation

TL;DR

This paper introduces a new Fourier spectral method for the cutoff Boltzmann equation, providing rigorous error estimates and validating its effectiveness through numerical experiments.

Contribution

It develops the first rigorous error analysis for a Fourier spectral scheme applied to the cutoff Boltzmann equation with Maxwellian and hard potentials.

Findings

Numerical experiments confirm the predicted accuracy of the method.

The scheme effectively captures the solution dynamics, including approach to equilibrium.

Abstract

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method for the equation with Maxwellian and hard potentials, (ii) the derivation of the first rigorous error estimates for the proposed schemes. Comprehensive numerical experiments validate the theory, confirming the predicted accuracy and illustrating the method's capability to capture solution dynamics, including the approach to equilibrium. The study thus provides a complete framework--from theoretical analysis to practical implementation--for the reliable computation of solutions to this foundational kinetic model.