A BOUNDARY LAYER PROBLEM FOR AN ASYMPTOTIC PRESERVING SCHEME IN THE QUASI-NEUTRAL LIMIT FOR THE EULER-POISSON SYSTEM

We consider the two-fluid Euler-Poisson system modeling the expansion of a quasi-neutral plasma in the gap between two electrodes. The plasma is injected from the cathode using boundary conditions which are not at the quasi-neutral equilibrium. This generates a boundary layer at the cathode. We numerically show that classical schemes as well as the asymptotic preserving scheme developed in [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208-234] are unstable for general Roe type solvers when the mesh does not resolve the small scale of the Debye length. We formally derive a model describing the boundary layer. Analyzing this problem, we determine well-adapted boundary conditions. These well-adapted boundary conditions stabilize general solvers without resolving the Debye length.