Numerical methods for the three-dimensional two-body problem in the action-at-a-distance electrodynamics

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses Sturmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies.