In high-temperature plasma physics, a strong magnetic field is usually used to confine charged particles. Therefore, for studying the classical mathematical models of the physical problems it needs to consider the effect of external magnetic fields. One of the important model equations in plasma is the Vlasov-Poisson equation with an external magnetic field. This equation usually has multi-scale characteristics and rich physical properties, thus it is very important and meaningful to construct numerical methods that can maintain the physical properties inherited by the original systems over long time. This paper extends the corresponding theory in Cartesian coordinates to general orthogonal curvilinear coordinates, and proves that a Poisson-bracket structure can still be obtained after applying the corresponding finite element discretization. However, the Hamiltonian systems in the new coordinate systems generally cannot be decomposed into sub-systems that can be solved accurately, so it is impossible to use the splitting methods to construct the corresponding geometric integrators. Therefore, this paper proposes a semi-implicit method for strong magnetic fields and analyzes the asymptotic stability of this method.

翻译：在高温度等离子体物理中，通常使用强磁场来约束带电粒子。因此，在研究物理问题的经典数学模型时，需要考虑外加磁场的影响。等离子体中重要的模型方程之一是带有外加磁场的Vlasov-Poisson方程。该方程通常具有多尺度特征和丰富的物理性质，因此构造能长时间保持原系统固有物理性质的数值方法具有重要意义。本文将笛卡尔坐标系下的相应理论推广到一般正交曲线坐标系，并证明在应用相应有限元离散后仍能获得泊松括号结构。然而，新坐标系下的哈密顿系统通常无法分解为可精确求解的子系统，因此无法使用分裂方法构造相应的几何积分器。为此，本文提出了一种适用于强磁场的半隐式方法，并分析了该方法的渐近稳定性。