A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

TL;DR

This paper introduces a deterministic spectral method for solving the Boltzmann equation using a Galerkin discretization with Hermite polynomial basis functions, demonstrating spectral convergence and applicability to low-speed rarefied gas flows.

Contribution

It presents a novel pseudo-spectral velocity space discretization method for the Boltzmann equation based on Hermite polynomials and Gauss-Hermite quadrature, enabling efficient numerical solutions.

Findings

Spectral convergence demonstrated in homogeneous relaxation tests.

Effective for simulating low-speed, slightly rarefied gas flows.

Applicable to two-dimensional cavity flow with varying Knudsen and Mach numbers.

Abstract

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows.