Numerical Methods for the Evolution of Fields with Applications to Plasmas

In this dissertation, we present a collection of algorithms for evolving fields in plasmas with applications to the Vlasov-Maxwell system. Maxwell's equations are reformulated in terms of the Lorenz and Coulomb gauge conditions to obtain systems involving wave equations. These wave equations are solved using the methods developed in this thesis and are combined with a particle-in-cell method to simulate plasmas. The particle-in-cell methods developed in this work treat particles using several approaches, including the standard Newton-Lorenz equations, as well as a generalized momentum formulation that eliminates the need to compute time derivatives of the field data. In the first part of this thesis, we develop and extend some earlier methods for scalar wave equations, which are used to update the potentials in these formulations. Our developments are based on a class of algorithms known as the MOL$^T$, which combines a dimensional splitting technique with a one-dimensional integral equation method. This results in methods that are unconditionally stable, can address geometry, and are $\mathcal{O}(N)$, where $N$ is the number of mesh points. Our work contributes methods to construct spatial derivatives of the potentials for this class of dimensionally-split algorithms, which are used to evolve particles.The second part of this thesis considers core algorithms used in the MOL$^T$ and the related class of successive convolution methods in the context of high-performance computing environments. We developed a novel domain decomposition approach that ultimately allows the method to be used on distributed memory computing platforms. Shared memory algorithms were developed using the Kokkos performance portability library, which permits a user to write a single code that can be executed on various computing devices with the architecture-dependent details being managed by the library. We optimized predominant loop structures in the code and developed a blocking pattern that prescribes parallelism at multiple levels and is also more cache-friendly. Moreover, the proposed iteration pattern is flexible enough to work with shared memory features available on GPU systems.The final part of this thesis presents the particle-in-cell method for the Vlasov-Maxwell system, which leverages the methods for fields and derivatives developed in this work. The proposed methods are applied to several test problems involving beams. Our results are generally encouraging and demonstrate the capabilities of the proposed field solvers in simulating basic plasma phenomena. Additionally, our results serve to validate the generalized momentum formulation, which will be the foundation of our future work.