An fully implicit scheme for numerical integration of the relativistic particle equation of motion

TL;DR

This paper introduces a fully implicit numerical scheme for solving the relativistic particle equation of motion, enabling accurate simulations of ultra-relativistic plasmas with Lorentz factors up to 10^9, surpassing previous limitations.

Contribution

The authors develop a novel fully implicit algorithm for relativistic particle dynamics in electromagnetic fields, improving accuracy and stability for high Lorentz factors in plasma simulations.

Findings

Efficiently simulates particles in constant electromagnetic fields.

Accurately follows electric drift motion at high Lorentz factors.

Performs well in variable electromagnetic field scenarios, matching analytical solutions.

Abstract

Relativistic strongly magnetized plasmas are produced in laboratories thanks to state-of-the-art laser technology but can naturally be found around compact objects such as neutron stars and black holes. Detailed studies of the behaviour of relativistic plasmas require accurate computations able to catch the full spatial and temporal dynamics of the system. Numerical simulations of ultra-relativistic plasmas face severe restrictions due to limitations in the maximum possible Lorentz factors that current algorithm can reproduce to good accuracy. In order to circumvent this flaw and repel the limit to $\gamma\approx10^9$, we design a new fully implicit scheme to solve the relativistic particle equation of motion in an external electromagnetic field using a three dimensional Cartesian geometry. We show some examples of numerical integrations in constant electromagnetic fields to prove the efficiency of our algorithm. The code is also able to follow the electric drift motion for high Lorentz factors. In the most general case of spatially and temporally varying electromagnetic fields, the code performs extremely well as shown by comparison with exact analytical solutions for the relativistic electrostatic Kepler problem as well as for linearly and circularly polarized plane waves.