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Particle-in-cell codes usually represent large groups of particles as asingle macroparticle. These codes are computationally efficient but loseinformation about the internal structure of the macroparticle. To improve theaccuracy of these codes, this work presents a method in which, as well astracking the macroparticle, the moments of the macroparticle are also tracked.Although the equations needed to track these moments are known, the coordinatetransformations for moments where the space and time coordinates are mixedcannot be calculated using the standard method for representing moments. Thesecoordinate transformations are important in astrophysical plasma, where thereis no preferred coordinate system. This work uses the language of Schwartzdistributions to calculate the coordinate transformations of moments. Both themoment tracking and coordinate transformation equations are tested by modellingthe motion of uncharged particles in a circular orbit around a black hole inboth Schwarzschild and Kruskal-Szekeres coordinates. Numerical testing showsthat the error in tracking moments is small, and scales quadratically. Thiserror can be improved by including higher order moments. By choosing anappropriate method for using these moments to deposit the charge back onto thegrid, a full particle-in-cell code can be developed.

Moment tracking and their coordinate transformations for macroparticles with an application to plasmas around black holes