Long term analysis of splitting methods for charged-particle dynamics

In this paper, we rigorously analyze the energy, momentum and magnetic moment behaviours of two splitting methods for solving charged-particle dynamics. The near-conservations of these invariants are given for the system under constant magnetic field or quadratic electric potential. By the approach named as backward error analysis, we derive the modified equations and modified invariants of the splitting methods and based on which, the near-conservations over long times are proved. Some numerical experiments are presented to demonstrate these long time behaviours.