Geometric particle-in-cell discretizations have been derived based on a discretization of the fields that is conforming with the de Rham structure of the Maxwell's equation and a standard particle-in-cell ansatz for the fields by deriving the equations of motion from a discrete action principle. While earlier work has focused on finite element discretization of the fields based on the theory of Finite Element Exterior Calculus, we propose in this article an alternative formulation of the field equations that is based on the ideas conveyed by mimetic finite differences. The needed duality being expressed by the use of staggered grids. We construct a finite difference formulation based on degrees of freedom defined as point values, edge, face and volume integrals on a primal and its dual grid. Compared to the finite element formulation no mass matrix inversion is involved in the formulation of the Maxwell solver. In numerical experiments, we verify the conservation properties of the novel method and study the influence of the various parameters in the discretization.

翻译：几何粒子网格离散化方法基于与麦克斯韦方程组de Rham结构相协调的场离散化以及通过离散作用量原理推导运动方程的标准粒子网格场假设而建立。尽管先前的研究主要集中于基于有限元外微积分理论的场有限元离散化，本文提出了一种基于拟有限差分思想的场方程替代表述。所需的对偶性通过交错网格的使用得以表达。我们构建了一种基于点值、边积分、面积分和体积分作为自由度的有限差分公式，这些自由度定义在原始网格及其对偶网格上。与有限元公式相比，该麦克斯韦求解器的表述不涉及质量矩阵求逆。在数值实验中，我们验证了新方法的守恒性质，并研究了离散化中各种参数的影响。