Numerical investigations of collisional physics have historically been impeded by the issue of computational expense. While the continuum-kinetic Vlasov-Maxwell-Fokker-Planck system is well-established in theory and has been used as the basis for many approximate fluid equations, simulations utilizing the distribution function are relatively uncommon, due primarily to the high dimensionality of the problem. However, advances in numerical methods are steadily making these models more accessible. In this work, we utilize the Gkeyll framework, which applies a novel, highly efficient discontinuous Galerkin (DG) finite element method to the Vlasov-Maxwell-Fokker-Planck system. We first investigate the Rayleigh-Taylor (RT) instability in a neutral gas in regimes of finite collisionality which are inaccessible to the fluid codes that are traditionally applied to this instability. Utilizing a spatially constant, finite collision frequency, we demonstrate the ability of the Vlasov-Boltzmann model to approach the fluid result at high collision frequency while also accessing a regime of intermediate collisionality in which the RT instability deviates greatly from classic fluid behavior. We then extend upon this finding by choosing a collision frequency that varies spatially, resulting in new dynamics with asymmetric diffusion affecting the development of the RT instability. Having demonstrated the utility of collisional kinetic modeling even in the simple case of a neutral gas with a basic collision operator, we transition to development and implementation of a fully-conservative, recovery-based DG algorithm for the full nonlinear Rosenbluth/Fokker-Planck collision operator (FPO). Details of the novel recovery scheme for the cross-derivatives and conservation enforcement are presented, and we show that the scheme converges and exhibits stability criteria as expected. Finally, the FPO is applied to test cases that demonstrate the importance of accurate handling of the velocity-dependent collision frequency as compared to an approximate model. / Doctor of Philosophy / Under the right conditions, the electrons and ions that comprise the particles in a gas separate, or ionize, forming a plasma. Plasma is the most common state of matter in the universe, existing at a wide range of scales. Whether concerning a supernova, the solar wind, a plume of material ablated by a laser, or a nuclear fusion reactor, all of these plasmas are governed by the same set of rules, with the main differences being which length and time scales are relevant. Understanding the dynamics of these collections of ionized particles offers a unique challenge, as particles interact not only by colliding with one another but through longer-range electromagnetic interactions. A number of methods exist for modeling plasmas, and one must choose which of the many scales in the plasma are relevant in order to make the best choice of model. In this work, we apply the continuum-kinetic method, which captures the statistical effect of individual particle motions while avoiding the noise that arises when tracking individual particles directly. Kinetic methods are not applied nearly as often as fluid methods, primarily because of the computational expense involved in resolving the wide range of scales and accounting for quantities that evolve as a function of both position and velocity. However, recent advances in numerical methods have made continuum-kinetic methods much more accessible. This work utilizes the Gkeyll code framework, which applies a discontinuous Galerkin method, to simulate plasma with a continuum-kinetic model. We begin by considering the Rayleigh-Taylor (RT) instability, which occurs when a heavy fluid is balanced atop a lighter fluid and perturbed, resulting in fluid mixing. The RT instability is ubiquitous in nature and is commonly modeled with fluid methods that assume particle collide with one another with effectively infinite frequency. With the continuum-kinetic method, we demonstrate that situations arise where the collision frequency is finite but the RT instability still grows. In these regimes, the instability growth is no longer well-described by fluid methods, and a kinetic model must be applied to accurately predict its evolution. Following this, we introduce an algorithm that utilizes a novel discontinuous Galerkin (DG) method to model one of the most complex and accurate collision operators for plasmas: the Fokker-Planck operator (FPO). The FPO is notoriously difficult to implement numerically and computationally expensive due to its nonlinear nature, so simulations generally utilize approximate forms rather than the full operator. By applying this DG method, we are able to ensure the numerical FPO implementation maintains many of the desirable properties of the original model while running highly efficiently. We conclude by verifying that the code is stable and highly accurate while reproducing expected results and improvements over simplified collision models.

A careful study of plasma-material interactions is essential to understand and improve the operation of devices where plasma contacts a wall such as plasma thrusters, fusion devices, spacecraft-environment interactions, to name a few. This work aims to advance our understanding of fundamental plasma processes pertaining to plasma-material interactions, sheath physics, and kinetic instabilities through theory and novel numerical simulations. Key contributions of this work include (i) novel continuum kinetic algorithms with novel boundary conditions that directly discretize the Vlasov/Boltzmann equation using the discontinuous Galerkin method, (ii) fundamental studies of plasma sheath physics with collisions, ionization, and physics-based wall emission, and (iii) theoretical and numerical studies of the linear growth and nonlinear saturation of the kinetic Weibel instability, including its role in plasma sheaths. The continuum kinetic algorithm has been shown to compare well with theoretical predictions of Landau damping of Langmuir waves and the two-stream instability. Benchmarks are also performed using the electromagnetic Weibel instability and excellent agreement is found between theory and simulation. The role of the electric field is significant during nonlinear saturation of the Weibel instability, something that was not noted in previous studies of the Weibel instability. For some plasma parameters, the electric field energy can approach magnitudes of the magnetic field energy during the nonlinear phase of the Weibel instability. A significant focus is put on understanding plasma sheath physics which is essential for studying plasma-material interactions. Initial simulations are performed using a baseline collisionless kinetic model to match classical sheath theory and the Bohm criterion. Following this, a collision operator and volumetric physics-based source terms are introduced and effects of heat flux are briefly discussed. Novel boundary conditions are developed and included in a general manner with the continuum kinetic algorithm for bounded plasma simulations. A physics-based wall emission model based on first principles from quantum mechanics is self-consistently implemented and demonstrated to significantly impact sheath physics. These are the first continuum kinetic simulations using self-consistent, wall emission boundary conditions with broad applicability across a variety of regimes. / Ph. D. / An understanding of plasma physics is vital for problems on a wide range of scales: from large astrophysical scales relevant to the formation of intergalactic magnetic fields, to scales relevant to solar wind and space weather, which poses a significant risk to Earth’s power grid, to design of fusion devices, which have the potential to meet terrestrial energy needs perpetually, and electric space propulsion for human deep space exploration. This work aims to further our fundamental understanding of plasma dynamics for applications with bounded plasmas. A comprehensive understanding of theory coupled with high-fidelity numerical simulations of fundamental plasma processes is necessary, this then can be used to improve improve the operation of plasma devices. There are two main thrusts of this work. The first thrust involves advancing the state-of-the-art in numerical modeling. Presently, numerical simulations in plasma physics are typically performed either using kinetic models such as particle-in-cell, where individual particles are tracked through a phase-space grid, or using fluid models, where reductions are performed from kinetic physics to arrive at continuum models that can be solved using well-developed numerical methods. The novelty of the numerical modeling is the ability to perform a complete kinetic calculation using a continuum description and evolving a complete distribution function in phase-space, thus resolving kinetic physics with continuum numerics. The second thrust, which is the main focus of this work, aims to advance our fundamental understanding of plasma-wall interactions as applicable to real engineering problems. The continuum kinetic numerical simulations are used to study plasma-material interactions and their effects on plasma sheaths. Plasma sheaths are regions of positive space charge formed everywhere that a plasma comes into contact with a solid surface; the charge inequality is created because mobile electrons can quickly exit the domain. A local electric field is self-consistently created which accelerates ions and retards electrons so the ion and electron fluxes are equalized. Even though sheath physics occurs on micro-scales, sheaths can have global consequences. The electric field accelerates ions towards the wall which can cause erosion of the material. Another consequence of plasma-wall interaction is the emission of electrons. Emitted electrons are accelerated back into the domain and can contribute to anomalous transport. The novel numerical method coupled with a unique implementation of electron emission from the wall is used to study plasma-wall interactions. While motivated by Hall thrusters, the applicability of the algorithms developed here extends to a number of other disciplines such as semiconductors, fusion research, and spacecraft-environment interactions.