On deterministic approximation of the Boltzmann equation in a bounded domain

TL;DR

This paper introduces a fully deterministic, spectrally accurate numerical method for solving the Boltzmann equation in bounded domains, capable of handling multi-scale rarefied gas flows efficiently.

Contribution

It presents a novel Fourier-based collision operator approximation combined with a finite volume scheme, enabling accurate and efficient simulations of rarefied gas dynamics.

Findings

Spectral accuracy in velocity space.

Computational cost of N log(N).

Effective for unsteady flows and hydrodynamic limits.

Abstract

In this paper we present a fully deterministic method for the numerical solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain for multi-scale problems. Periodic, specular reflection and diffusive boundary conditions are discussed and investigated numerically. The collision operator is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity with a computational cost of $N\,\log(N)$, where $N$ is the number of degree of freedom in velocity space. This algorithm is coupled with a second order finite volume scheme in space and a time discretization allowing to deal for rarefied regimes as well as their hydrodynamic limit. Finally, several numerical tests illustrate the efficiency and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects, trend to equilibrium).