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.

设计并分析了准中性标度下欧拉-泊松（EP）系统的渐近守恒（AP）和能量稳定格式。在质量和动量的对流通量以及静电势梯度中引入适当的稳定项，从而导致机械能的耗散，从而导致溶液的熵稳定性。时间离散本质上是半隐式的，而空间离散使用标记和单元（MAC）网格上的有限体积框架。全离散格式的数值解析涉及两个步骤：求解势能的线性椭圆问题以及密度和速度的显式评估。所提出的方案具有几个物理相关属性，例如密度正性、熵稳定性以及与连续 EP 系统弱公式的一致性。论证了该方案的AP性质，即网格参数相对于德拜长度的有界性及其与拟中性极限系统的一致性。数值案例研究的结果证实了所提出方法的稳健性和效率。