正交曲线坐标系下磁化Vlasov--Poisson方程的渐近稳定Particle-in-Cell方法
Asymptotically stable Particle-in-Cell methods for the magnetized Vlasov--Poisson equations in orthogonal curvilinear coordinates
AI总结：
针对带外磁场的Vlasov-Poisson方程，本文将笛卡尔坐标系下的相关理论推广到正交曲线坐标系，提出强磁场下的半隐式Particle-in-Cell方法并证明其渐近稳定性。
摘要：
在高温等离子体物理中，通常使用强磁场来约束带电粒子。因此，研究这类物理问题的经典数学模型时需要考虑外磁场的影响。等离子体中的重要模型方程之一是带外磁场的Vlasov-Poisson方程。该方程通常具有多尺度特性和丰富的物理性质，因此构建能够长时间保持原系统固有物理性质的数值方法十分重要且有意义。本文将笛卡尔坐标系下的相关理论推广到一般正交曲线坐标系，证明了采用相应的有限元离散后仍可得到泊松括号结构。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建相应的几何积分器。为此，本文针对强磁场提出了一种半隐式方法，并分析了该方法的渐近稳定性。
英文摘要：
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.
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