Explicit volume-preserving numerical schemes for relativistic trajectories and spin dynamics

TL;DR

This paper introduces explicit, energy- and volume-preserving numerical schemes for simulating relativistic particle trajectories and spin dynamics, demonstrating improved long-term accuracy over traditional methods like Boris pusher.

Contribution

The paper develops a new class of explicit numerical schemes based on Clifford algebra that conserve energy and volume, with second order convergence, for relativistic particle and spin dynamics.

Findings

Numerical schemes are energy- and volume-conserving and second order accurate.

Error remains bounded in long-term simulations, outperforming Boris pusher.

Spin dynamics in plane wave fields are effectively captured.

Abstract

A class of explicit numerical schemes is developed to solve for the relativistic dynamics and spin of particles in electromagnetic fields, using the Lorentz-BMT equation formulated in the Clifford algebra representation of Baylis. It is demonstrated that these numerical methods, reminiscent of the leapfrog and Verlet methods, share a number of important properties: they are energy-conserving, volume-conserving and second order convergent. These properties are analysed empirically by benchmarking against known analytical solutions in constant uniform electrodynamic fields. It is demonstrated that the numerical error in a constant magnetic field remains bounded for long time simulations in contrast to the Boris pusher, whose angular error increases linearly with time. Finally, the intricate spin dynamics of a particle is investigated in a plane wave field configuration.