  A class of explicit numerical schemes is developed to solve for the
relativistic dynamics and spin of particles in electromagnetic fields, using
the Lorentz-BMT equation formulated in the Clifford algebra representation of
Baylis. It is demonstrated that these numerical methods, reminiscent of the
leapfrog and Verlet methods, share a number of important properties: they are
energy-conserving, volume-conserving and second order convergent. These
properties are analysed empirically by benchmarking against known analytical
solutions in constant uniform electrodynamic fields. It is demonstrated that
the numerical error in a constant magnetic field remains bounded for long time
simulations in contrast to the Boris pusher, whose angular error increases
linearly with time. Finally, the intricate spin dynamics of a particle is
investigated in a plane wave field configuration.
