We present convergent gravitational waveforms extracted from three-dimensional, numerical simulations in the wave zone and with causally disconnected boundaries. These waveforms last for multiple periods and are very accurate, showing a peak error to peak amplitude ratio of 2% or better. Our approach includes defining the Weyl scalar Psi_4 in terms of a three-plus-one decomposition of the Einstein equations; applying, for the first time, a novel algorithm due to Misner for computing spherical harmonic components of our wave data; and using fixed mesh refinement to focus resolution on non-linear sources while simultaneously resolving the wave zone and maintaining a causally disconnected computational boundary. We apply our techniques to a (linear) Teukolsky wave, and then to an equal mass, head-on collision of two black holes. We argue both for the quality of our results and for the value of these problems as standard test cases for wave extraction techniques.
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