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