We develop a structure-preserving numerical discretization for the electrostatic Euler-Poisson equations with a constant magnetic field. The scheme preserves positivity of the density, positivity of the internal energy and a minimum principle of the specific entropy, as well as global properties, such as total energy balance. The scheme uses an operator splitting approach composed of two subsystems: the compressible Euler equations of gas dynamics and a source system. The source system couples the electrostatic potential, momentum, and Lorentz force, thus incorporating electrostatic plasma and cyclotron motions. Because of the high-frequency phenomena it describes, the source system is discretized with an implicit time-stepping scheme. We use a PDE Schur complement approach for the numerical approximation of the solution of the source system. Therefore, it reduces to a single non-symmetric Poisson-like problem that is solved for each time step. Our focus with the present work is on the efficient solution of problems close to the magnetic-drift limit. Such asymptotic limit is characterized by the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales that differ by over 10 orders of magnitude, making their numerical solution quite challenging. We illustrate the capability of the scheme by computing a diocotron instability and present growth rates that compare favorably with existing analytical results. The model, though a simplified version of the Euler-Maxwell's system, represents a stepping stone toward electromagnetic solvers that are capable of working in the electrostatic and magnetic-drift limits, as well as the hydrodynamic regime.

翻译：本文针对含恒定磁场的静电欧拉-泊松方程，提出了一种结构保持的数值离散格式。该格式保持了密度正性、内能正性及比熵的最小值原理，同时保持了全局特性如总能量平衡。本方案采用算子分裂方法，将系统分解为两个子系统：气体动力学的可压缩欧拉方程与源项系统。源项系统耦合了静电势、动量与洛伦兹力，从而包含了静电等离子体振荡与回旋运动。由于该系统描述的是高频物理现象，我们采用隐式时间推进格式进行离散。通过偏微分方程舒尔补方法对源项系统进行数值求解，将其简化为每个时间步需要求解的单个非对称类泊松问题。本研究的重点在于高效求解接近磁漂移极限的问题。该渐近极限的特征在于缓慢运动的平滑流与极高频率振荡共存，其时间尺度差异超过10个数量级，使得数值求解极具挑战性。我们通过计算双极漂移不稳定性展示了该格式的性能，其增长率与现有解析结果高度吻合。该模型虽为欧拉-麦克斯韦方程组的简化版本，但为构建能够同时处理静电极限、磁漂移极限及流体动力学区域的电磁求解器奠定了重要基础。