Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Abstract

We present a numerical algorithm for evaluating the Boltzmann collision operator with \begin{document}$O(N^2)$\end{document} 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. Split and non-split forms of the collision operator are considered, which are forms of the collision operator that have separate and simultaneous evaluations of the gain and loss terms, respectively. Smaller numerical errors are observed in the conserved quantities in simulations using the non-split form.