MULTIPOLE EXPANSIONS FOR THE FOKKER-PLANCK-LANDAU OPERATOR

We give a sequence of operators approximating the Fokker-Planck-Landau collision operator. This sequence is obtained by aplying the fast mul tipole method based on the work by Greengard and Rokhlin [17], and ten ds to the exact Fokker-Planck-Landau operator with an arbitrary accura cy. These operators satisfy the physical properties such as the conser vation of mass, momentum, energy and the decay of the entropy. Further more, the quadratic structure due to the velocity coupling in the expr ession of the Fokker-Planck-Landau operator is removed in the approxim ating operators. This fact reduces seriously the computationnal cost o f numerical simulations of the Fokker-Planck-Landau equation. Finally, we give numerical conservative and entropy discretizations solving th e homogeneous Fokker-Planck-Landau equation using the fast multipole m ethod. In addition to the deterministic character of these approximati ons, they give satisfactory results in terms of accuracy and CPU time.