A FAST PETROV-GALERKIN SPECTRAL METHOD FOR THE MULTIDIMENSIONAL BOLTZMANN EQUATION USING MAPPED CHEBYSHEV FUNCTIONS

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 Rd. 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 nonuniform fast Fourier transform. We demonstrate the superior accuracy of the proposed method in comparison to the Fourier spectral method through a series of two-dimensional and three-dimensional examples.

Automated summary

In this paper, a Petrov-Galerkin spectral method for the Boltzmann equation in unbounded domain is introduced. The method utilizes carefully chosen mapped Chebyshev functions as basis functions to achieve desired convergence and conservation properties. The proposed method is shown to have superior accuracy compared to the Fourier spectral method through a series of two-dimensional and three-dimensional examples.