Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator

In this paper we show that the use of spectral-Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with innite-order accuracy momentum and energy. Consistency of the method is also proved and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coeÆcients associated to the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions conrm the advantages of the method.