Multipole expansions for the Fokker-Planck-Landau operator

Summary.

We give a sequence of operators approximating the Fokker-Planck-Landau collision operator. This sequence is obtained by aplying the fast multipole method based on the work by Greengard and Rokhlin [17], and tends to the exact Fokker-Planck-Landau operator with an arbitrary accuracy. These operators satisfy the physical properties such as the conservation of mass, momentum, energy and the decay of the entropy. Furthermore, the quadratic structure due to the velocity coupling in the expression of the Fokker-Planck-Landau operator is removed in the approximating operators. This fact reduces seriously the computationnal cost of numerical simulations of the Fokker-Planck-Landau equation. Finally, we give numerical conservative and entropy discretizations solving the homogeneous Fokker-Planck-Landau equation using the fast multipole method. In addition to the deterministic character of these approximations, they give satisfactory results in terms of accuracy and CPU time.