A fast spectral method for the Boltzmann collision operator with general collision kernels

TL;DR

This paper introduces a new fast spectral method for the Boltzmann collision operator that significantly reduces computational complexity and memory requirements, applicable to general collision kernels unlike previous methods.

Contribution

The paper presents a novel spectral method with lower complexity and broader applicability to various collision kernels compared to existing approaches.

Findings

Complexity reduced to O(MN^4 log N) from O(N^6)

No precomputation needed for variable hard sphere model

Applicable to arbitrary collision kernels with numerical validation

Abstract

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discretization points in each of the three velocity dimensions and $M$ is the total number of discretization points on the sphere and $M\ll N^2$. Furthermore, it requires no precomputation for the variable hard sphere (VHS) model and only $O(MN^4)$ memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available \cite{MP06} (with complexity $O(MN^3\log N)$), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.