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

Abstract

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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad hoc</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">de-Rham</i> sequences and lays a groundwork for understanding conditions that must be satisfied for consistency. Several numerical results demonstrate the applicability of these rubrics.