  In this paper we present a new variable time step criterion for the
velocity-Verlet algorithm allowing to correctly simulate the dynamics of
charged particles exchanging energy via Coulomb collisions while minimising
simulation time. We present physical arguments supporting the use of the
criterion along with numerical results proving its validity. We numerically
show that $\bar{\textrm{H}}^{+}$ ions with 18 meV initial energy can be
captured and sympathetically cooled by a Coulomb crystal of $\textrm{Be}^{+}$
and $\textrm{HD}^{+}$ in less than 10 ms, an important result for the GBAR
project.
