Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation

TL;DR

This paper introduces fast summation methods for discrete-velocity approximations of the Boltzmann equation, significantly reducing computational costs while maintaining accuracy.

Contribution

It develops new algorithms that decrease the complexity of evaluating the Boltzmann collision operator in discrete-velocity schemes.

Findings

Reduced computational complexity from O(N^{2d+1}) to O( N^d log N )

Almost no loss of accuracy in the new methods

Applicable to high-dimensional velocity space computations

Abstract

Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically $O(N^{2d+1})$ where $d$ is the dimension of the velocity space. In this paper, following the ideas introduced in [27,28], we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from $O(N^{2d+1})$ to $O(\bar{N}^d N^d\log_2 N)$, $\bar{N} << N$, with almost no loss of accuracy.