The anatomy of Boris type solvers and the Lie operator formalism for deriving large time-step magnetic field integrators

TL;DR

This paper uses Lie operator formalism to analyze Boris solvers, revealing new on-orbit integrators that improve large time-step accuracy in magnetic field simulations, and clarifies historical misconceptions about solver behaviors.

Contribution

It introduces a novel on-orbit Boris solver derived via Lie operators, distinct from finite-difference methods, and clarifies the relationship between existing schemes and Buneman's cycloid fitting.

Findings

Identified two methods to eliminate trajectory errors in Boris solvers.

Discovered a new on-orbit solver not derivable from finite-difference schemes.

Showed the symmetric second-order Boris solver's superior accuracy at large time-steps.

Abstract

This work gives a Lie operator derivation of various Boris solvers via a detailed study of trajectory errors in a constant magnetic field. These errors in the gyrocenter location and the gyroradius are the foundational basis for why Boris solvers existed, independent of any finite-difference schemes. This work shows that there are two distinct ways of eliminating these errors so that the trajectory of a charged particle in a constant magnetic field is exactly on the cyclotron orbit. One way reproduces the known second-order symmetric Boris solver. The other yields a previously unknown, but also on-orbit solver, not derivable from finite-difference schemes. By revisiting some historical calculations, it is found that many publications do not distinguish the poorly behaved leap-frog Boris solver from the symmetric second-order Boris algorithm. This symmetric second-order Boris solver's trajectory is much more accurate and remains close to the exact orbit in a combined $nonuniform$ electric and magnetic field at time-steps greater than the cyclotron period. Finally, this operator formalism showed that Buneman's cycloid fitting scheme is mathematically identical to Boris' on-orbit solver and that Boris' E-B splitting is unnecessary.