Pushing particles in extreme fields

The update of the particle momentum in an electromagnetic simulation typically employs the Boris scheme, which has the advantage that the magnetic field strictly performs no work on the particle. In an extreme field, however, it is found that onerously small time steps are required to maintain accuracy. One reason for this is that the operator splitting scheme fails. In particular, even if the electric field impulse and magnetic field rotation are computed exactly, a large error remains. The problem can be analyzed for the case of constant, but arbitrarily polarized and independent electric and magnetic fields. The error can be expressed in terms of exponentials of nested commutators of the generators of boosts and rotations. To second order in the field, the Boris scheme causes the error to vanish, but to third order in the field, there is an error that has to be controlled by decreasing the time step. This paper introduces a scheme that avoids this problem entirely, while respecting the property that magnetic fields cannot change the particle energy.