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

$\bar{\textrm{H}}^{+}$ Sympathetic Cooling Simulations with a Variable Time Step