  An asymptotic preserving and energy stable scheme for the Euler-Poisson
system under the quasineutral scaling is designed and analysed. Correction
terms are introduced in the convective fluxes and the electrostatic potential,
which lead to the dissipation of mechanical energy and the entropy stability.
The resolution of the semi-implicit in time finite volume in space
fully-discrete scheme involves two steps: the solution of an 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
the entropy stability and the consistency with the weak formulation of the
continuous Euler-Poisson system. The AP property of the scheme, i.e. the
boundedness of the mesh parameters with respect to the Debye length and its
consistency with the quasineutral limit system, is shown. The results of
numerical case studies are presented to substantiate the robustness and
efficiency of the proposed method.

Numerical Analysis;Numerical Analysis