Rubrics for Charge Conserving Current Mapping in Finite Element Electromagnetic Particle in Cell Methods

Modeling of kinetic plasmas using electromagnetic particle in cell (EM-PIC) methods is a well worn problem, in that methods developed have been used extensively both understanding physics and exploiting them for device design. EM-PIC tools have largely relied on finite difference methods coupled with particle representations of the distribution function. Refinements to ensure consistency and charge conservation have largely been ad hoc efforts specific to finite difference methods. Meanwhile, solution methods for field solver have grown by leaps and bounds with significant performance metrics compared to finite difference methods. Developing new EM-PIC computational schemes that leverage modern field solver technology means re-examining analysis framework necessary for self-consistent EM-PIC solution. In this article, we prescribe general rubrics for charge conservation, demonstrate how these are satisfied in conventional finite difference PIC as well as finite element PIC, and prescribe a novel charge conserving finite element PIC. Our effort leverages proper mappings on to de-Rham sequences and lays a groundwork for understanding conditions that must be satisfied for consistency. Several numerical results demonstrate the applicability of these rubrics.

I. Introduction

Modeling plasmas and other charged media as sources is a needed capability to better understand and design many real-world devices. Of the many techniques to model this phenomena, the particle-in-cell (PIC) method has been a popular approach. A wide variety of fields depend on PIC as a computational method, ranging from fundamental science of space weather, nuclear fusion, atomic processes, on one hand, to high technology applications like particle accelerators, coherent electromagnetic radiation sources, and plasma processing devices on the other. PIC, at a glance, works as follows: charged media, represented as a collection of particles, move in a domain under the influence of fields that are due to the motion of these particles. This calls for self-consistent solution of Maxwell’s equations in conjunction with equation of motion. Given application drivers stated earlier, there has been extensive development of EM-PIC methods [1]–[5]. Several of these have been used extensively in the design of devices [6]–[10]. While traditional application were electrically large albeit geometrically simple, there is burgeoning interest in devices with significantly geometric complexity but smaller footprint. As a result, there is a need for EM-PIC methods with greater capabilities [11], [12].