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 quasineutral 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 [9] 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. Analysing this problem, we determine well-adapted boundary conditions. These well-adapted boundary conditions stabilize general solvers without resolving the Debye length.