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 app roximation of the Boltzmann equation in the velocity space permits one to o btain spectrally accurate numerical solutions at a reduced computational co st. We prove that the spectral algorithm preserves the total mass and appro ximates with infinite-order accuracy momentum and energy. Consistency of th e method is also proved, and a stability result for a smoothed positive sch eme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in so me cases can be computed explicitly. Numerical examples for homogeneous tes t problems in two and three dimensions confirm the advantages of the method .