An asymptotic preserving and energy stable scheme for the Euler-Poisson system in the quasineutral limit

Abstract

An asymptotic preserving (AP) and energy stable scheme for the Euler-Poisson (EP) system under the quasineutral scaling is designed and analysed. Appropriate stabilisation terms are introduced in the convective fluxes of mass and momenta, and the gradient of the electrostatic potential which lead to the dissipation of mechanical energy and consequently the entropy stability of solutions. The time discretisation is semi-implicit in nature, whereas the space discretisation uses a finite volume framework on a marker and cell (MAC) grid. The numerical resolution of the fully-discrete scheme involves two steps: the solution of a linear elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, entropy stability and the consistency with the weak formulation of the continuous EP system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to Debye length and its consistency with the quasineutral limit system, is demonstrated. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.