  We present fully general relativistic simulations of the quasi-circular
inspiral and merger of charged, non-spinning, binary black holes with
charge-to-mass ratio $\lambda \le 0.3$. We discuss the key features that
enabled long term and stable evolutions of these binaries. We also present a
formalism for computing the angular momentum carried away by electromagnetic
waves, and the electromagnetic contribution to black-hole horizon properties.
We implement our formalism and present the results for the first time in
numerical-relativity simulations. In addition, we compare our full non-linear
solutions with existing approximate models for the inspiral and ringdown
phases. We show that Newtonian models based on the quadrupole approximation
have errors of 20 % - 100 % in key gauge-invariant quantities. On the other
hand, for the systems considered, we find that estimates of the remnant black
hole spin based on the motion of test particles in Kerr-Newman spacetimes agree
with our non-linear calculations to within a few percent. Finally, we discuss
the prospects for detecting black hole charge by future gravitational-wave
detectors using either the inspiral-merger-ringdown signal or the ringdown
signal alone.
