Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

TL;DR

This paper presents a numerical method for solving high-order regularized moment equations derived from the Boltzmann-BGK equation, emphasizing efficiency and flexibility without explicitly formulating the moment equations.

Contribution

The method directly solves the Boltzmann equation using a numerical approach for Grad's moments and regularization, avoiding explicit moment equations and improving computational efficiency.

Findings

Validated convergence with different numbers of moments.

Demonstrated efficiency in flux calculations.

Provided numerical examples for 1D and 2D problems.

Abstract

We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not need explicitly the moment equations. As an instead, we directly start from the Boltzmann equation and perform Grad's moment method \cite{Grad} and the regularization technique \cite{Struchtrup2003} numerically. We define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions. For the collision term, the BGK model is adopted so that the production step can be done trivially based on the Hermite expansion. Extensive numerical examples for one- and two-dimensional problems are presented. Convergence in moments can be validated by the numerical results for different number of moments.