几何Fourier粒子网格法
On Geometric Fourier Particle In Cell Methods
摘要：
在这篇文章中，我们描述了一个统一的框架变分电磁粒子计划的频谱类型，我们提出了一种新的频谱粒子在细胞（PIC）计划，保持离散的哈密顿结构。我们的工作是基于一个新的抽象变分推导的粒子计划，建立在一个德拉姆复杂的低拉格朗日离散使用粒子近似的分布函数。在这个框架中，它扩展了最近的基于有限元的几何电磁PIC（GEMPIC）方法的各种场求解器，电磁势和场的离散化表示的de Rham序列的兼容空间，和粒子场耦合过程中所描述的近似算子，与微分算子的序列交换。特别是，对于谱麦克斯韦求解器的截断$L^2$投影的选择，使用连续的傅里叶变换系数的交换近似算子产生无网格的粒子在傅里叶方法，而频谱粒子网格法通过使用从网格计算的离散傅里叶变换系数获得。通过引入一系列新的谱拟微分近似算子，我们得到了一种新的具有离散Hamilton结构的变分谱PIC方法，我们称之为Fourier-GEMPIC。完全离散的计划，然后使用哈密顿分裂过程，导致明确的时间步长，保持高斯定律和离散泊松括号与哈密顿结构。这些显式步骤与标准谱PIC方法有许多相似之处。由于在我们的框架中允许任意滤波器，我们还讨论了混叠误差，并研究了一个自然的反向滤波过程，以减轻抗混叠平滑粒子形状所造成的阻尼。
英文摘要：
In this article we describe a unifying framework for variational electromagnetic particle schemes of spectral type, and we propose a novel spectral Particle-In-Cell (PIC) scheme that preserves a discrete Hamiltonian structure. Our work is based on a new abstract variational derivation of particle schemes which builds on a de Rham complex where Low's Lagrangian is discretized using a particle approximation of the distribution function. In this framework, which extends the recent Finite Element based Geometric Electromagnetic PIC (GEMPIC) method to a variety of field solvers, the discretization of the electromagnetic potentials and fields is represented by a de Rham sequence of compatible spaces, and the particle-field coupling procedure is described by approximation operators that commute with the differential operators in the sequence. In particular, for spectral Maxwell solvers the choice of truncated $L^2$ projections using continuous Fourier transform coefficients for the commuting approximation operators yields the gridless Particle-in-Fourier method, whereas spectral Particle-in-Cell methods are obtained by using discrete Fourier transform coefficients computed from a grid. By introducing a new sequence of spectral pseudo-differential approximation operators, we then obtain a novel variational spectral PIC method with discrete Hamiltonian structure that we call Fourier-GEMPIC. Fully discrete schemes are then derived using a Hamiltonian splitting procedure, leading to explicit time steps that preserve the Gauss laws and the discrete Poisson bracket associated with the Hamiltonian structure. These explicit steps share many similarities with standard spectral PIC methods. As arbitrary filters are allowed in our framework, we also discuss aliasing errors and study a natural back-filtering procedure to mitigate the damping caused by anti-aliasing smoothing particle shapes.
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