  Time-centered, hence second-order, methods for integrating the relativistic
momentum of charged particles in an electromagnetic field are derived. A new
method is found by averaging the momentum before use in the magnetic rotation
term, and an implementation is presented that differs from the relativistic
Boris Push [1] only in the method for calculating the Lorentz factor. This is
shown to have the same second-order accuracy in time as that (Boris Push) [1]
found by splitting the electric acceleration and magnetic rotation and that [2]
found by averaging the velocity in the magnetic rotation term. All three
methods are shown to conserve energy when there is no electric field. The Boris
method and the current method are shown to be volume-preserving, while the
method of [2] and the current method preserve the $\vec{E} \times \vec{B}$
velocity. Thus, of these second-order relativistic momentum integrations, only
the integrator introduced here both preserves volume and gives the correct
$\vec{E} \times \vec{B}$ velocity. While all methods have error that is
second-order in time, they deviate from each other by terms that increase as
the motion becomes relativistic. Numerical results show that [2] develops
energy errors near resonant orbits of a test problem that neither
volume-preserving integrator does.
  [1] J. Boris, Relativistic plasma simulation-optimization of a hybrid code,
in: Proc. Fourth Conf. Num. Sim. Plasmas, Naval Res. Lab, Wash. DC, 1970, pp.
3-67.
  [2] J.-L. Vay, Simulation of beams or plasmas crossing at relativistic
velocity, Physics of Plasmas (1994-present) 15 (5) (2008) 056701.
