We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative entropy functionals. For this kind of models including porous media equations, Fokker-Planck equations for plasma physics or dumbbell models for polymer flows, it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme is built from a discretization of the steady equation and preserves steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After proving well-posedness, stability, exponential return to equilibrium and convergence, we present several numerical results which confirm the accuracy and underline the efficiency to preserve large-time asymptotic.

Abstract. We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio me/m i tends to zero.Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satisfies an anisotropic drift-diffusion equation. To achieve this task, we overcome some specific technical issues of our model such as the strong effect of the magnetic field on electrons and the lack of regularity at the limit. With methods including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model.