  We study the convergence properties of our implementation of the 'moving
punctures' approach at very high resolutions for an equal-mass, non-spinning,
black-hole binary. We find convergence of the Hamiltonian constraint on the
horizons and the L2 norm of the Hamiltonian constraint in the bulk for sixth
and eighth-order finite difference implementations.
  The momentum constraint is more sensitive, and its L2 norm shows clear
convergence for a system with consistent sixth-order finite differencing, while
the momentum and BSSN constraints on the horizons show convergence for both
sixth and eighth-order systems. We analyze the gravitational waveform error
from the late-inspiral, merger, and ringdown. We find that using several
lower-order techniques for increasing the speed of numerical relativity
simulations actually lead to apparently non-convergent errors. Even when using
standard high-accuracy techniques, rather than seeing clean convergence, where
the waveform phase is a monotonic function of grid resolution, we find that the
phase tends to oscillate with resolution, possibly due to stochastic errors
induced by grid refinement boundaries. Our results seem to indicate that one
can obtain gravitational waveform phases to within 0.05 rad. (and possibly as
small as 0.015 rad.), while the amplitude error can be reduced to 0.1%. We then
compare with the waveforms obtained using the cZ4 formalism. We find that the
cZ4 waveforms have larger truncation errors for a given resolution, but the
Richardson extrapolation phase of the cZ4 and BSSN waveforms agree to within
0.01 rad., even during the ringdown.
