Numerical approximation of collisional plasmas by high-order methods
The evolution of a collisional plasma constituted of different species of particles is commonly described by the Fokker-Planck-Landau equation at the kinetic level. It describes binary collisions between charged particles with long-range Coulomb interactions. Many different deterministic numerical schemes have been considered to Fokker-Planck type equations. Most of these methods have proven their efficiency in the homogeneous case, but few results are available in the nonhomogeneous situation. The main goal of this paper is to develop a scheme in the position depending case and in the whole 3D velocity space. The first step consists to construct a good approximation of the Vlasov equation. A scheme is proposed, using a phase space grid. Space and velocity derivatives are approximated by a centered finite volume method. An adapted approximation of the electric field allows to obtain a numerical scheme that conserves the total energy. As this kind of discretization does not ensure the positivity of the unknown distribution function, slope correctors are introduced. The distribution function is reconstructed following the second order PFC method. When the slope correctors act the total energy is not conserved anymore. Finally, the collision operator is approximated in the case of interaction of particles of different species. A multigrid method is used to reduce the computational cost. Several numerical results are presented to illustrate the efficiency of the method. In the context of the Landau damping, the results here are in very good agreement with theoretical results in the literature. For strong perturbations the effect of collisions on the damping of the electric energy and on the long time behavior of the solution is observed.
scientific article; zbMATH DE number 2126400