A dual grid geometric electromagnetic particle in cell method

TL;DR

This paper introduces a novel dual grid geometric electromagnetic particle-in-cell method based on mimetic finite differences, which preserves conservation properties without requiring mass matrix inversion, offering an alternative to finite element approaches.

Contribution

It proposes a dual grid finite difference formulation for electromagnetic PIC that avoids mass matrix inversion and leverages mimetic finite difference ideas with staggered grids.

Findings

Conservation properties are verified numerically.

The method's parameters influence discretization accuracy.

The approach offers an alternative to finite element discretizations.

Abstract

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.