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

Abstract

Modeling of kinetic plasmas using electromagnetic particle in cell methods (EM-PIC) is a problem that is well worn, 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 an 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 paper, 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.

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

“…We follow the usual representation where ρ(t, r) = q Np p=1 δ(r − r p (t)) and J(t, r) = q Np p=1 v p (t)δ(r − r p (t)). Generalization to other shape function (instead of delta functions) is certainly possible 32 , but is beyond the scope of this paper and not pertinent to the central thesis of this paper. As in PIC schemes, the particles are moved using the Lorentz force F(t, r) = q(E(t, r) + v(t, r) × B(t, r)) and Newton's equations.…”

“…The left hand side of this equation is the discrete Laplacian, and coefficients Ēn ns are the appropriate values of the potential due to the charge density on the right hand side at that instance of time. Further note, as shown in 32 this equation can be rewritten as,…”

“…As an aside, in the rest of the paper, we will use the term conservation laws to denote these equations and admit to an abuse of terminology! This problem was solved relatively recently 31,32 with results demonstrating satisfaction of conservation laws to almost machine precision. But this is not really the end of the story.…”

Higher Order Charge Conserving Electromagnetic Finite Element Particle in Cell Method

“…where r p (t) and v p (t) are the position and velocity of particle p. In this work the shape functions are chosen to be Dirac delta functions, though generalization to other shape functions is possible [6]. The particular choice shape function is immaterial to the results of this paper.…”

“…Ref. [6] rigorously develops the conditions that should be satisfied, and demonstrates how current EM-PIC formulations satisfy these conditions. For instance, the spatial and temporal basis sets used in an explicit FDTD timestepping scheme, together with an appropriate integration of the path, satisfies these constraints.…”