High order volume-preserving algorithms for relativistic charged   particles in general electromagnetic fields
  We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. Via expanding the phase space to include time $t$, we give a more general construction of volume-preserving methods that can be applied to systems with time-dependent electromagnetic fields. The newly derived methods provide numerical solutions with good accuracy and conservative properties over long time of simulation. Furthermore, because of the use of processing technique the high order methods are explicit, and cost less than the methods derived from standard compositions, thus are more efficient. The results are verified by the numerical experiments. Linear stability analysis of the methods show that the high order processed method allows larger time step size during integration. 
Yang He, Yajuan Sun, Ruili Zhang, Yulei Wang, Jian Liu, and Hong Qin
physics.comp-ph physics.plasm-ph