Fast Spectral Methods for the Fokker–Planck–Landau Collision Operator

In this paper we present a new spectral method for the fast evaluation of the Fokker–Planck–Landau (FPL) collision operator. The method allows us to obtain spectrally accurate numerical solutions with simply O(n log2n) operations in contrast with the usual O(n2) cost of a deterministic scheme. We show that the method preserves the total mass whereas momentum and energy are approximated with spectral accuracy. Numerical results for the FPL equation for Maxwell molecules and for Coulomb interactions in two and three dimensions in velocity space are also given.