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

TL;DR

This paper introduces a method for tracking moments of macroparticles and their coordinate transformations in particle-in-cell simulations, enhancing accuracy for astrophysical plasmas around black holes.

Contribution

It presents a novel approach using Schwartz distributions to compute coordinate transformations of moments, improving particle-in-cell modeling in curved spacetime.

Findings

Error in moment tracking scales quadratically with time

Coordinate transformations are accurately computed using Schwartz distributions

Method demonstrated with particles orbiting black holes in different coordinates

Abstract

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 work presents a method in which, as well as tracking the macroparticle, the moments of the macroparticle are also tracked. Although the equations needed to track these moments are known, the coordinate transformations for moments where the space and time coordinates are mixed cannot be calculated using the standard method for representing moments. These coordinate transformations are important in astrophysical plasma, where there is no preferred coordinate system. This work uses the language of Schwartz distributions to calculate the coordinate transformations of moments. Both the moment tracking and coordinate transformation equations are tested by modelling the motion of uncharged particles in a circular orbit around a black hole in both Schwarzschild and Kruskal-Szekeres coordinates. Numerical testing shows that the error in tracking moments is small, and scales quadratically. This error can be improved by including higher order moments. By choosing an appropriate method for using these moments to deposit the charge back onto the grid, a full particle-in-cell code can be developed.