A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation

Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new Particle-in-Cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509--3537]. It treats all of the stages of the standard PIC update - charge deposition, force interpolation, the field solve, and the particle push - with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one- and two- dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4th-order method can achieve comparable accuracy to the 2nd-order method with many fewer resolution elements.Comment: 18 pages, 10 figures, submitted to SIS

A 4th-order particle-in-cell method with phase-space remapping for the vlasov-poisson equation

Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new particle-in- cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. H. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509-3537]. It treats all of the stages of the standard PIC update|charge deposition, force interpolation, the field solve, and the particle push|with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one- and two- dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4th- order method can achieve accuracy comparable to that of the 2nd-order method and with considerably fewer resolution elements

Deterministic-Kinetic Computational Analyses of Expansion Flows and Current-Carrying Plasmas

Spacecraft electric propulsion (EP) takes advantage of the ability of electric and magnetic fields to accelerate plasmas to high velocities to generate efficient thrust. The thermionic hollow cathode is a critical component to both gridded-ion and Hall-effect thrusters, the state-of-the-art devices of the EP discipline. However, experiments demonstrate that the hollow cathode is plagued by erosion of its surfaces by the plasma, which may eventually cause premature failure of the device. This erosion has been linked to the ion-acoustic instability (IAI), a kinetic plasma instability which operates in the cathode plume. Existence of this kinetic instability has prevented numerical simulation from predicting the operating characteristics and lifetime of the hollow cathode device. Therefore, this thesis utilizes deterministic-kinetic (DK) simulation of gas and plasma flows to further the understanding of the IAI as it relates to the hollow cathode plume and to ultimately develop a predictive hollow cathode simulation platform. Towards these goals, two approaches to applying the DK simulation method to the hollow cathode plasma are undertaken: hybrid-kinetic simulation and fully-kinetic simulation. Hybrid-kinetic simulations utilize a kinetic description of the heavy propellant particles while using a reduced-order, fluid approach for the light electrons. Two unique two-dimensional, axisymmetric kinetic schemes are developed, one for neutral particles and one for ions; the schemes are verified by comparison with solutions obtained using the direct-simulation Monte Carlo method and with an analytic solution for a rarefied neutral jet flow. Assuming quasi-neutrality in the hollow cathode plasma and using the Boltzmann relation for the plasma potential, the hybrid-kinetic solver is applied to the problem of NASA's NSTAR discharge hollow cathode. Partial validation is achieved through agreement with experimental Langmuir probe data in the near-orifice region, while shortcomings of the solver such as use of a simplified electron model are discussed. Fully-kinetic simulations, where all species are considered kinetically, are carried out to study the IAI. The anomalous resistivity generated by the IAI is measured from one-dimensional fully-kinetic simulations and compared with a closure model commonly used in hollow cathode fluid codes, finding that the agreement with the closure model varies based on simulation domain size and electron Mach number. Further, the formation of high-energy tails in the ion velocity distribution function is observed near the transition to the Buneman instability, another instability of current-carrying plasmas. Two-dimensional kinetic simulations of current-carrying instabilities are carried out, finding that the nature of nonlinear saturation of the IAI differs significantly from that shown in one-dimensional simulations. A phenomenon known as the off-axis instability generates waves propagating normal to the current direction which eventually reach energy levels close to that of the waves along the current direction. Further fully-kinetic simulations demonstrate the formation of weak plasma double layers, regions of plasma which sustain a potential gradient, in the nonlinear saturation stage of the IAI. These double layers are found to be ubiquitous in all plasma species considered, even the heavy xenon ions commonly used in hollow cathodes. Phase space analysis suggests the double layers form from ion-acoustic wave packets which grow into ion phase space holes. Spectral analysis demonstrates a shift towards smaller wavenumbers which marks this transition. An electron two-stream instability is spawned due to the potential well of the double layer, where spectral analyses demonstrate that a simple theoretical expression well-predicts the resulting wave phase velocity.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169806/1/vazsonyi_1.pd

Drift instabilities, anomalous transport, and heating in low-temperature plasmas

Plasma is an ideal gas of charged particles (ions and electrons) in addition to neutral particles. The presence of charged particles results in the generation of electric and magnetic fields that serve as the primary mechanism of the interaction and coupling of particles. As a result, various nonlinear collective phenomena occur in the plasma, the understanding of many of which remains elusive today. On the other hand, plasmas have many applications in different branches of science and technology. Different kinds of plasmas are studied in the atmospheric and space sciences. In the semiconductor industry, the fabrication of electronic chips relies heavily on plasma etching. Plasma is used in modern electrical thrusters for producing the driving force of satellites and spacecrafts. It is also used in future fusion reactors for producing abundant clean energy. Therefore, understanding the complicated phenomena in plasma is important for predicting and controlling its behaviours in various conditions. In this regard, nonlinear phenomena, such as turbulence, are formidable barriers to understanding plasma behaviours. These phenomena are described by nonlinear differential equations that can be barely understood by analytical means and are usually investigated by numerical simulations. Because of this, it is also important to understand the effect of numerical artifacts on simulations. In this thesis, we investigate the nonlinear characteristics of drift instabilities and the role of numerical methods in our understanding of these instabilities. The drift instabilities are driven by excess free energy that exists due to the average (drift) velocities of electron and ion components in plasmas. As a result of these instabilities, the amplitude of fluctuations grows while the drift energy converts into electrostatic energy. This growth continues until the nonlinear effects, such as turbulence, trapping, and wave-wave interactions, become active. As a result of these nonlinear effects, the growth of the fluctuations saturates. In this thesis, our focus will be on two particular types of drift instabilities, namely the Buneman instability and electron-cyclotron drift instability (ECDI). The Buneman instability is driven when a beam of electrons is injected into the stationary ions, while both electrons and ions are unmagnetized. In the ECDI, however, the electrons are magnetized and are also influenced by an external electric field, perpendicular to the magnetic field. This configuration of fields leads to the E × B drift of the electrons that drives the ECDI. Many kinetic simulations are performed, and several nonlinear phenomena such as trapping, heating, anomalous transport, backward waves, and transition of magnetized plasmas to the unmagnetized regime are studied with regard to both instabilities. For the study of the nonlinear effects of drift instabilities, a grid-based Vlasov code is developed and used. The numerical method used in this code is the “semi-Lagrangian” method, which is among the most popular methods for continuum simulations of plasma. In the study of the drift instabilities, we compare the results of the semi-Lagrangian Vlasov simulations with the more traditional particle-in-cell (PIC) method. The results of these benchmarking studies reveal several similarities and discrepancies between Vlasov and particle-in-cell simulations, showing how the numerical methods can interfere with the physics of the problems