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G. Kitzler, J. Schöberl:
"Efficient Spectral Methods for the spatially homogeneous Boltzmann equation";
in: "ASC Report 13/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 26.

We present a spectral Petrov-Galerkin method for the spatially homogeneous Boltzmann equation. We approximate the density distribution function by high order multivariate Lagrange polynomials
in Gauss Hermite points, multiplied by a Gaussian peak with adjusted mean and width; the test functions are polynomials. Our focus is on an efficient scheme for applying the Boltzmann collision
operator. The first improvement is to transform the collision integral to mean and relative velocity which allows to use cheap numerical integration rules for the first one. The second improvement is a fast transformation from Lagrange via Hermite to a hierarchical basis in Polar coordinates. In this basis, the innermost integral operator becomes diagonal. We conclude with a numerical example
demonstrating the achieved speed up.

http://www.asc.tuwien.ac.at/preprint/2013/asc13x2013.pdf