# Structure-preserving second-order integration of relativistic charged particle trajectories in electromagnetic fields

## TL;DR

This paper introduces a new second-order, structure-preserving integration method for relativistic charged particle trajectories in electromagnetic fields, improving energy conservation and velocity accuracy over existing methods.

## Contribution

A novel second-order integration method that preserves volume and the correct  imes \u00B5 B velocity, differing from the relativistic Boris Push by its Lorentz factor calculation.

## Key findings

- The new method conserves energy in absence of electric fields.
- It preserves volume and  imes B5 B velocity, unlike some existing methods.
- Numerical tests show improved energy stability near resonant orbits.

## Abstract

Time-centered, hence second-order, methods for integrating the relativistic momentum of charged particles in an electromagnetic field are derived. A new method is found by averaging the momentum before use in the magnetic rotation term, and an implementation is presented that differs from the relativistic Boris Push [1] only in the method for calculating the Lorentz factor. This is shown to have the same second-order accuracy in time as that (Boris Push) [1] found by splitting the electric acceleration and magnetic rotation and that [2] found by averaging the velocity in the magnetic rotation term. All three methods are shown to conserve energy when there is no electric field. The Boris method and the current method are shown to be volume-preserving, while the method of [2] and the current method preserve the $\vec{E} \times \vec{B}$ velocity. Thus, of these second-order relativistic momentum integrations, only the integrator introduced here both preserves volume and gives the correct $\vec{E} \times \vec{B}$ velocity. While all methods have error that is second-order in time, they deviate from each other by terms that increase as the motion becomes relativistic. Numerical results show that [2] develops energy errors near resonant orbits of a test problem that neither volume-preserving integrator does. [1] J. Boris, Relativistic plasma simulation-optimization of a hybrid code, in: Proc. Fourth Conf. Num. Sim. Plasmas, Naval Res. Lab, Wash. DC, 1970, pp. 3-67. [2] J.-L. Vay, Simulation of beams or plasmas crossing at relativistic velocity, Physics of Plasmas (1994-present) 15 (5) (2008) 056701.