This thesis presents a new Particle-in-Cell method for numerically simulating plasmas under the Vlasov-Maxwell system. Maxwell's equations and the Newton-Lorentz force are both recast in terms of vector and scalar potentials under the Lorenz gauge. This results in a set of decoupled wave equations governing the potentials and a Hamiltonian system with a generalized momentum formulation governing the particles. The Particle-in-Cell framework for solving a plasma system requires two main components, a method for updating the fields and a method for updating the particles of the system.The first part of this thesis introduces the Method of Lines Transpose, or MOLT, as a way of solving partial differential equations in general and the wave equation in particular. Additionally it introduces a new particle pusher, the Improved Asymmetrical Euler Method, that is a modification of a previously existing method. We deploy these two techniques in the Particle-in-Cell framework. In this section in particular MOLT employs a dimensional splitting algorithm, solving a set of one dimensional boundary value problems using a Green's function. This will all be done using one particular temporal discretization scheme, the first order Backward Difference Formula. Numerical results are shown to give evidence for the quality of these techniques, though it is noteworthy that the combination of this wave solver and particle pusher does not satisfy the Lorenz gauge condition, nor does it satisfy the involutions of Maxwell's equations, otherwise known as Gauss's laws.The second part of this thesis fills this lacuna, suggesting two ways for doing so. First it will consider theory to connect satisfaction of the continuity equation with satisfaction of the Lorenz gauge. It will consider in particular a way of satisfying this theory with multi-dimensional Green's functions, eschewing the dimension splitting of the first part. It will additionally consider the solution of the boundary value problems via other numerical techniques such as the Fast Fourier Transform or Finite Difference approach, ultimately choosing these for simplicity. The second approach will consider a gauge correction technique. It will be shown that both of these preserve the gauge, but the first method will additionally satisfy the involutions of Maxwell's equations. In a similar manner to the first part, it will do so using the first order Backward Difference Formula as the temporal discretization scheme. Numerical evidence will be given to support the theory developed.The third part of this thesis will generalize the theory connecting the satisfaction of the continuity equation with satisfaction of the Lorenz gauge and, in most cases, with Gauss's Laws. It will extend this theory to not only all orders of the Backward Difference Formulation, but to a family of second order time centered methods, arbitrary stage diagonally implicit Runge-Kutta methods, and all orders of Adams-Bashforth methods. In all but the diagonally implicit Runge-Kutta methods, Gauss's laws will be shown to be satisfied if the Lorenz gauge is. Once again numerical evidence will be given to support this.Finally some future projects will be suggested to capitalize on this work.
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