High-Order Particle Integration for Particle-In-Cell Schemes using Boris with Spectral Deferred Corrections

The study of plasmas plays an important role in both science and technology. Plasma dynamics can be found wherever charged particles or materials interact with and generate electromagnetic fields, covering more orders of magnitude in scale and density than any other type of matter. Plasma phenomena dominate the dynamics of the sun, stars and space between them and are important to a variety of technologies, from fusion reactors to spacecraft propulsion. As plasma behaviour and associated mathematical relations are naturally complex, numerical methods and computer simulation play a crucial role in furthering the field. Particle-in-Cell (PIC) is a class of numerical scheme currently used in the simulation of hot diffuse plasmas, or denser plasmas at small scales. One crucial part of such schemes is the particle integrator, which solves the particle equations of motion, typically via time discretisation of the Newton-Lorentz force. For nearly forty years, the dominant algorithm for charged particle tracking has been leapfrog integration using Boris' algorithm. The combined scheme is often referred to simply as the classic Boris integrator and provides a directly computable centre-difference discretisation for the implicit system. As the Boris algorithm is intrinsically second order accurate, a tunable order algorithm based on Boris and spectral deferred corrections (Boris-SDC) was recently proposed and demonstrated to exhibit high order time convergence for a single-particle Penning trap with exactly known electromagnetic fields. The faster reduction in error as time-step size is decreased allowed Boris-SDC to be more computationally efficient than classic Boris, but whether the advantageous characteristics of Boris-SDC would extend to PIC and approximated fields was not investigated. This thesis contributes the implementation and performance testing of Boris-SDC within PIC schemes and generalises Boris-SDC to the relativistic regime. This relativistic extension to Boris-SDC is shown to retain higher order time convergence and improved computational performance when compared to classic Boris even in highly relativistic regimes ($>99\%$ speed of light). The relativistic Boris-SDC integrator is shown to produce less unphysical drift than classic Boris in the force-free scenario where electric and magnetic forces cancel. The algorithmic modifications required to implement Boris-SDC within PIC are highlighted first and the impact of spatial electric field approximation on particle integrator performance is demonstrated for the electrostatic case (ESPIC). The relativistic Boris-SDC integrator is then derived and implemented within the open-source PIC code Runko, demonstrating capability of Boris-SDC to work with existing codes. Finally, performance tests are conducted in the form of work-precision comparisons to classic Boris for two electrostatic benchmarks, the two-stream instability and Landau damping. The spatial field approximation inherent to ESPIC imposes an error saturation on the time convergence of the particle integrator, inversely proportional to the spatial resolution, which limits the achievable global error. The limited accuracy is found to erode the computational performance of Boris-SDC, as the low level of error required to offset the added computational cost of SDC cannot be reached. Above the spatial saturation point however, Boris-SDC is found to retain high order time convergence and higher accuracy for a fixed time-step size than classic Boris. As a final note, suggestions are given for further work, including use of higher order spatial methods, investigation on the significance of momentum error vs. spatial error as well as PIC applications wherein Boris-SDC might be useful despite the lack of a clear performance gain.