We develop, test and compare new numerical and geometrical methods for improving the accuracy of extracting waveforms using characteristic evolution. The new numerical method involves use of circular boundaries to the stereographic grid patches which
cover the spherical cross-sections of the outgoing null cones. We show how an angular version of numerical dissipation can be introduced into the characteristic code to damp the high frequency error arising form the irregular way the circular patch boundary cuts through the grid. The new geometric method involves use of the Weyl tensor component $Psi_4$ to extract the waveform as opposed to the original approach via the Bondi news function. We develop the necessary analytic and computational formula to compute the $O(1/r)$ radiative part of $Psi_4$ in terms of a conformally compactified treatment of null infinity. These methods are compared and calibrated in test problems based upon linearized waves.
We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes whic
h provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.
