The relativistic Vlasov-Maxwell system (RVM) models the behavior of collisionless plasma, where electrons and ions interact via the electromagnetic fields they generate. In the RVM system, electrons could accelerate to significant fractions of the speed of light. An idea that is actively being pursued by several research groups around the globe is to accelerate electrons to relativistic speeds by hitting a plasma with an intense laser beam. As the laser beam passes through the plasma it creates plasma wakes, much like a ship passing through water, which can trap electrons and push them to relativistic speeds. Such setups are known as laser wakefield accelerators, and have the potential to yield particle accelerators that are significantly smaller than those currently in use. Ultimately, the goal of such research is to harness the resulting electron beams to generate electromagnetic waves that can be used in medical imaging applications.

High-order accurate numerical discretizations of kinetic Vlasov plasma models are very effective at yielding low-noise plasma simulations, but are computationally expensive to solve because of the high dimensionality. In addition to the general difficulties inherent to numerically simulating Vlasov models, the relativistic Vlasov-Maxwell system has unique challenges not present in the non-relativistic case. One such issue is that operator splitting of the phase gradient leads to potential instabilities, thus we require an alternative to operator splitting of the phase.

The goal of the current work is to develop a new class of high-order accurate numerical methods for solving kinetic Vlasov models of plasma. The main discretization in configuration space is handled via a high-order finite element method called the discontinuous Galerkin method (DG). One difficulty is that standard explicit time-stepping methods for DG suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. The maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work, we overcome this difficulty by introducing a novel time-stepping strategy: the regionally-implicit discontinuous Galerkin (RIDG) method. The RIDG is method is based on an extension of the Lax-Wendroff DG (LxW-DG) method, which previously had been shown to be equivalent (for linear constant coefficient problems) to a predictor-corrector approach, where the prediction is computed by a space-time DG method (STDG). The corrector is an explicit method that uses the space-time reconstructed solution from the predictor step. In this work, we modify the predictor to include not just local information, but also neighboring information. With this modification, we show that the stability is greatly enhanced; we show that we can remove the polynomial degree dependence of the maximum time-step and show vastly improved time-steps in multiple spatial dimensions. Upon the development of the general RIDG method, we apply it to the non-relativistic 1D1V Vlasov-Poisson equations and the relativistic 1D2V Vlasov-Maxwell equations. For each we validate the high-order method on several test cases. In the final test case, we demonstrate the ability of the method to simulate the acceleration of electrons to relativistic speeds in a simplified test case.