Evaluating High Order Discontinuous Galerkin Discretization of the Boltzmann Collision Integral in $O(N^2)$ Operations Using the Discrete Fourier Transform

TL;DR

This paper introduces an efficient $O(N^2)$ algorithm for evaluating the Boltzmann collision operator using high order discontinuous Galerkin discretizations and the discrete Fourier transform, significantly improving computational speed.

Contribution

The paper develops a novel $O(N^2)$ algorithm for the Boltzmann collision operator leveraging Fourier transforms and Galerkin discretizations, enhancing computational efficiency.

Findings

Significant speedup over direct evaluation methods.

Reduced numerical error in conserved quantities with simultaneous gain and loss evaluation.

Validated approach in the spatially homogeneous case.

Abstract

We present a numerical algorithm for evaluating the Boltzmann collision operator with $O(N^2)$ operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Simultaneous and separate evaluations of the gain and loss terms of the collision operator were considered. Less numerical error was observed in the conserved quantities with simultaneous evaluation.