Positive, conservative, equilibrium state preserving and implicit difference schemes for the isotropic Fokker-Planck-Landau equation

The aim of this paper is to describe the discretization of the Fokker-Planck-Landau (FPL) collision term in the isotropic case, which models the self-collision for the electrons when they are totally isotropized by heavy particles background such as ions. The discussion focuses on schemes, which could preserve positivity, mass, energy and Maxwellian equilibrium. The Chang and Cooper method is widely used by plasma's physicists for the FPL (and for Fokker-Planck type equations). We present a new variant that is both positive and conservative contrary to the existing one's. In the category of others difference schemes we propose a simple scheme on non-uniform grid, which is both positive, conservative and equilibrium state preserving in opposition to what exists. The case of Coulombian potentials is emphasized. We address also the problem of the time discretization. In particular we show how to recast some implicit methods to get band diagonal system and to solve it by direct method with a linear cost. Numerical tests are performed.