Approximation of the Boltzmann Collision Operator Based on Hermite Spectral Method

TL;DR

This paper introduces a Hermite spectral Galerkin method for the Boltzmann equation, providing an efficient way to approximate collision operators with high accuracy, especially for low-order moments.

Contribution

It develops a new spectral method based on Hermite expansion and proposes an algorithm for accurate coefficient evaluation and collision model construction.

Findings

Efficiently captures low-order moments of the distribution.

Provides a practical algorithm for coefficient computation.

Demonstrates high accuracy in numerical experiments.

Abstract

Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inverse-power-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.