Particle-in-cell codes usually represent large groups of particles as a single macroparticle. These codes are computationally efficient but lose information about the internal structure of the macroparticle. To improve the accuracy of these codes, this thesis presents a method in which, as well as tracking the macroparticle, the moments of the macroparticle are also tracked.
One representation of moments uses integrals. In this representation of moments, the moment tracking equations are known, but the coordinate transformations for moments where the space and time coordinates are mixed cannot be calculated. These coordinate transformations are important in astrophysical plasma, where there is no preferred coordinate system. An alternative representation of moments uses Schwartz distributions. By using the language of Schwartz distributions, the equations to track the moments, and perform coordinate transformations of moments are calculated. The moment tracking and coordinate transformation equations are tested by modelling the motion of uncharged particles in a circular orbit around a black hole. Numerical testing shows that the error in tracking moments is small, and scales quadratically.
Two different methods to find the current distribution from a set of moments are presented. The first reconstructs the original distribution function used to find the moments, and derives the current distribution from the reconstructed distribution function. The second method uses the language of Schwartz distributions to directly calculate the current from the set of moments. The current distribution construction equations are tested for a variety of distribution functions, and show that using the language of Schwartz distributions introduces errors, but is computationally faster. The error in moment tracking, coordinate transformations, and in finding the current can be improved by including higher order moments.
The considerations needed to create a full particle-in-cell code, and how this code can be evaluated, are discussed.
