  In this work, we propose a new Galerkin-Petrov method for the numerical
solution of the classical spatially homogeneous Boltzmann equation. This method
is based on an approximation of the distribution function by associated
Laguerre polynomials and spherical harmonics and test an a variational manner
with globally defined three-dimensional polynomials. A numerical realization of
the algorithm is presented. The algorithmic developments are illustrated with
the help of several numerical tests.

Numerical Analysis