Error and long-term analysis of two-step symmetric methods for relativistic charged-particle dynamics

In this work, we consider the error estimates and the long-time conservation or near-conservation of geometric structures, including energy, mass shell and phase-space volume, for four two-step symmetric methods applied to relativistic charged-particle dynamics. We begin by introducing a two-step symmetric numerical method based on a splitting scheme that exactly preserves the mass shell and the phase-space volume of the relativistic system. Building on this formulation, we develop three additional two-step symmetric methods with further modifications, for which the long-time near-conservation of energy and mass shell can be rigorously established through the backward error analysis. All methods are shown to achieve second-order accuracy. The theoretical results are illustrated and complemented by numerical experiments.