Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties

TL;DR

This paper proves that the Fourier-Galerkin spectral method converges spectrally for the Boltzmann equation with uncertainties in collision kernel and initial conditions, extending known results to stochastic settings.

Contribution

It establishes the spectral convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties, using new spaces and norms that incorporate velocity and random variables.

Findings

Spectral convergence holds for the Boltzmann equation with uncertainties.

New analytical spaces and norms are developed for high-regularity variables.

Provides a foundation for convergence analysis of fully discretized stochastic systems.

Abstract

It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.