A Boundary Layer Problem for an Asymptotic Preserving Scheme in the Quasi-Neutral Limit for the Euler–Poisson System

Abstract

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.

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

“…A second direction of further investigation would be to look at problems in which hydrodynamic and diffusive regimes are present simultaneously in different parts of the spatial domain. Many efforts have been done for (semi-)implicit asymptotic-preserving schemes (some of them cited in the introduction); a number of these techniques (such as an a priori modeling of boundary layers [21,53]) can be readily applied in conjunction with the projective integration method proposed here. In a similar way, this approach can be applied to the region R 2,PRK , where we expand τ (θ) as: τ (θ) = C 1 (θ)z 1/K + C 2 (θ)z 2/K + h.o.t.…”

On Hydrodynamic Models for LEO Spacecraft Charging

“…The numerical treatment of this kind of asymptotic problem leads to severe stiff problems, which require a specific treatment. A deep understanding of the boundary layer formation and of the scale separation helps in designing an efficient numerical scheme, as in [26].…”

Rigorous derivation of the compressible Navier-Stokes equations from the two-fluid Navier-Stokes-Maxwell equations

“…For more results on asymptotic limits with small parameters for one-fluid flows, see [2,18]. For the asymptotic limit problem in a bounded domain, see [3,4,21,23,25] and the references therein.…”