  We present results from a new code for binary black hole evolutions using the
moving-puncture approach, implementing finite differences in generalised
coordinates, and allowing the spacetime to be covered with multiple
communicating non-singular coordinate patches. Here we consider a regular
Cartesian near zone, with adapted spherical grids covering the wave zone. The
efficiencies resulting from the use of adapted coordinates allow us to maintain
sufficient grid resolution to an artificial outer boundary location which is
causally disconnected from the measurement. For the well-studied test-case of
the inspiral of an equal-mass non-spinning binary (evolved for more than 8
orbits before merger), we determine the phase and amplitude to numerical
accuracies better than 0.010% and 0.090% during inspiral, respectively, and
0.003% and 0.153% during merger. The waveforms, including the resolved higher
harmonics, are convergent and can be consistently extrapolated to $r\to\infty$
throughout the simulation, including the merger and ringdown. Ringdown
frequencies for these modes (to $(\ell,m)=(6,6)$) match perturbative
calculations to within 0.01%, providing a strong confirmation that the remnant
settles to a Kerr black hole with irreducible mass $M_{\rm irr} =
0.884355\pm20\times10^{-6}$ and spin $S_f/M_f^2 = 0.686923 \pm 10\times10^{-6}$
