Simulation of space charge and plasma is an integral part of many scientific and engineering processes, with specific applications of pulsed power, particle accelerators, integrated chip manufacture, satellites, and medicine, to name a few examples. A number of methods have been used to simulate the underlying physics. One such method, Particle-In-Cell (PIC), does this by mapping all particle-field interactions through a mesh, and using macro-particles as statistically significant markers on the distribution function phase space. While PIC has been around since the 1960 much of the work has been focused on Finite Difference Time Domain (FDTD) for solving electromagnetic fields. Recently there has been a push for using Finite Element Methods due to ease at which it can model complex geometry. The challenge with using FEM or any method with an unstructured grid is developing a consistent methodology for evolving fields and particle populations such that all conservation laws are held. Developing this scheme is the principal contribution of this thesis. Specifically, to set the stage for the development, we start with developing a set of three dimensional validation tests for PIC methods. These tests have either an analytic or quasi-analytic solutions. Next, we delve into structure preserving geometric algorithms that inform how different quantities that appear both Maxwell’s equations and Newton’s laws should be represented. Using these, we prescribe a consistent particle mapping methodology for an implicit field solver. We prove and demonstrate that conservation laws are satisfied. Finally, to extend the regime of applicability of this approach, we develop quasi-Helmholtz projectors that ensures analytical satisfaction of the equation of continuity and Gauss’ law for magnetic flux density, and discrete satisfaction of Gauss’ law for electric flux density. The efficacy of these methods and their benefits are presented on benchmark examples developed earlier.
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