An entropy scheme for the Fokker - Planckcollision operator of plasma kinetic

We propose a nite diierence scheme to approximate the Fokker-Planck collision operator in 3 velocity dimensions. The principal feature of this scheme is to provide a decay of the numerical entropy. As a consequence, it preserves the collisional invariants and its stationary solutions are the discrete Maxwellians. We consider both the whole velocity-space problem and the bounded velocity problem. In the latter case, we provide artiicial boundary conditions which preserve the decay of the entropy. The Fokker-Planck equation is used for the description of binary collisions between charged particles, for which the interaction potential is the long-range Coulomb interaction. By opposition to short-range interactions which come into play in collisions between neutral molecules, the long-range Coulomb interaction cannot be described by a Boltzmann like operator, mainly because it gives rise to divergent integrals. This diiculty is dealt with by cutting-oo the in-tegrals and retaining the principal part of the Boltzmann operator when the cut-oo parameter tends to zero; this leads to the Fokker-Planck operator (see 8],,11] for a physical presentation and 1], 4], 5], 6] for mathematical proofs of the convergence of the Boltzmann operator to the Fokker-Planck one in various situations). The cut-oo parameter is identiied as one of the most important parameters characterizing a plasma: the plasma parameter g = (nn 3 D) ?1 ; which is the inverse of the number of particles within a Debye sphere (n is the density of the plasma and D is the Debye length) 5]. From a theoretical viewpoint, the