Polynomial Finite-Size Shape Functions for Electromagnetic Particle-in-Cell Algorithms Based on Unstructured Meshes

In this article, we describe a novel implementation of finite-size (super) particles described by polynomial-based spatial shape factors in electromagnetic particle-in-cell (EM-PIC) algorithms for kinetic plasma simulations based on unstructured meshes. The proposed implementation is aimed at mitigating (spurious) numerical Cherenkov radiation effects while preserving exact charge conservation. This is achieved by employing a representation of Maxwell's equations based on the exterior calculus of differential forms and a discretization of twisted differential forms associated with the primal mesh by Whitney forms, while ordinary differential forms are associated with the dual mesh. The set of full-discrete Maxwell's equations is then obtained by using a finite-element Galerkin method and a leap-frog time integration, leading to a mixed D-H finite-element time-domain Maxwell field solver. In the proposed EM-PIC implementation, the gather and scatter steps of the algorithm yield integrals of polynomials over polyhedral domains in 3-D space (or polygonal domains in 2-D space) that can be evaluated analytically. We provide numerical examples confirming charge conservation down to machine precision on an unstructured mesh.