A fast Petrov-Galerkin spectral method for the multi-dimensional Boltzmann equation using mapped Chebyshev functions

TL;DR

This paper introduces a fast Petrov-Galerkin spectral method using mapped Chebyshev functions for the Boltzmann equation in unbounded domains, achieving high accuracy and conservation properties with efficient computation.

Contribution

The paper develops a novel Petrov-Galerkin spectral method with mapped Chebyshev functions for unbounded domains, enabling fast algorithms via NUFFT and improved accuracy over Fourier methods.

Findings

Demonstrates superior accuracy in 2D and 3D examples.

Achieves conservation properties in the numerical scheme.

Provides a fast computational algorithm using NUFFT.

Abstract

Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this method requires a domain truncation which is unphysical since the collision operator is defined in $\mathbb{R}^d$. In this paper, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we are able to construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT). We demonstrate the superior accuracy of the proposed method in comparison to the Fourier spectral method through a series of 2D and 3D examples.