The anatomy of Boris type solvers and large time-step plasma simulations

This work gives a Lie operator derivation of various Boris solvers via a detailed study of first and second-order trajectory errors in a constant magnetic field. These errors in the gyro-circle center and gyro-radius are the foundational basis for why Boris solvers existed, independent of any finite-difference schemes. The elimination of these errors then forces the second-order solver's trajectory to be exactly on the gyro-circle. By revisiting some historical calculations, it is found that many publications do not distinguish the poorly behaved first-order leap-frog solver with the correct second-order Boris algorithm. This work shows that this second-order Boris solver is much more accurate then previously thought and that its trajectory remains close to the exact orbit in a combined $nonuniform$ electric and magnetic field at time-steps greater than the cyclotron period.