Burnett Spectral Method for the Spatially Homogeneous Boltzmann Equation

TL;DR

This paper introduces a spectral method using Burnett polynomials for the spatially homogeneous Boltzmann equation, significantly reducing computational costs while maintaining accuracy.

Contribution

It presents a novel spectral approach leveraging Burnett polynomials and sparsity to improve efficiency in solving the Boltzmann equation.

Findings

Reduces computational cost by one to two orders of magnitude.

Demonstrates high accuracy and efficiency through numerical examples.

Seamlessly integrates with BGK-type models.

Abstract

We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules. The proposed method can couple seamlessly with the BGK-type modelling techniques to make future applications affordable. The implementation of the algorithm is discussed in detail, including a numerical scheme to compute all the coefficients accurately, and the design of the data structure to achieve high cache hit ratio. Numerical examples are provided to demonstrate the accuracy and efficiency of our method