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

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 E × B velocity. Thus, of these second-order relativistic momentum integrations, only the integrator introduced here both preserves volume and gives the correct E × 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.

A Comprehensive Comparison of Relativistic Particle Integrators

“…Volume preservation is discussed thoroughly in Higuera & Cary (2017), concluding that the Boris scheme and the Higuera-Cary scheme are volume preserving but the Vay scheme is not. Volume preservation is defined here as the preservation of the differential volume, which is preserved by any solution of the underlying differential equation, with a finite time step (Higuera & Cary 2017). We test the preservation of the gyroradius in several different cases of magnetic and electric fields in Section 3.1.…”

“…As shown by Vay (2008), this decoupling leads to the breaking of the Lorentz invariance and the introduction of spurious forces for the Boris method. The Vay method and the HC method are proven to maintain their Lorentz invariance (Vay 2008;Higuera & Cary 2017). The implicit algorithm does not decouple the electric and magnetic field advance, avoiding the problem of maintaining Lorentz invariance completely, as demonstrated in Lapenta & Markidis (2011).…”

“…It is expected that the Boris scheme will introduce a small phase lag of order t 2 D ( ) at each time step. The HC scheme will introduce a smaller phase lag of order t 3 D ( ) (Higuera & Cary 2017). Figure 3 shows the path followed by the particle during the gyration and the conservation of γ.…”

“…There are several numerical methods to treat relativistic particle motion accurately. Here, we aim to test a selection of available classical and recent explicit leapfrog methods (Boris 1970;Vay 2008;Higuera & Cary 2017). We also present and test a newly implemented fully implicit relativistic method that conserves energy exactly.…”

A fully implicit numerical integration of the relativistic particle equation of motion

“…Boris algorithm has been extended to relativistic motion by Vay (2008). A volume-preserving relativistic integrator was recently developed by Higuera & Cary (2017). Splitting techniques for high order symmetric volume-preserving methods can be found in He, Yang et al (2016).…”