Conservative and entropy numerical scheme for the isotropic Fokker-Planck-Landau equation

Homogeneous Fokker-Planck-Landau equation is investigated for coulombian potential and isotropic distribution function i.e. when the distribution function depends only on time and on the modulus of the velocity. We derive a new conservative and entropy decaying semi-discretized Landau equation for which we prove the existence of global in time positive solutions. For the time-discretized equation, we give lower bound for the time step which guaranteed positivity and entropy decaying of the numerical solution.