We develop and calibrate a characteristic waveform extraction tool whose major improvements and corrections of prior versions allow satisfaction of the accuracy standards required for advanced LIGO data analysis. The extraction tool uses a characteristic evolution code to propagate numerical data on an inner worldtube supplied by a 3 þ 1 Cauchy evolution to obtain the gravitational waveform at null infinity. With the new extraction tool, high accuracy and convergence of the numerical error can be demonstrated for an inspiral and merger of mass M binary black holes even for an extraction worldtube radius as small as R ¼ 20M. The tool provides a means for unambiguous comparison between waveforms generated by evolution codes based upon different formulations of the Einstein equations and based upon different numerical approximations.

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 É 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 Oð1=rÞ radiative part of É 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 which 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.

Computational methods are essential to provide waveforms from coalescing black holes, which are expected to produce strong signals for the gravitational wave observatories being developed. Although partial simulations of the coalescence have been reported, scientifically useful waveforms have so far not been delivered. The goal of the AppleswithApples (AwA) Alliance is to design, coordinate and document standardized code tests for comparing numerical relativity codes. The first round of AwA tests have now being completed and the results are being analyzed. These initial tests are based upon periodic boundary conditions designed to isolate performance of the main evolution code. Here we describe and carry out an additional test with periodic boundary conditions which deals with an essential feature of the black hole excision problem, namely a nonvanishing shift. The test is a shifted version of the existing AwA gauge wave test. We show how a shift introduces an exponentially growing instability which violates the constraints of a standard harmonic formulation of Einstein's equations. We analyze the Cauchy problem in a harmonic gauge and discuss particular options for suppressing instabilities in the gauge wave tests. We implement these techniques in a finite difference evolution algorithm and present test results. Although our application here is limited to a model problem, the techniques should benefit the simulation of black holes using harmonic evolution codes. PACS numbers: PACS number(s): 04.20Ex, 04.25Dm, 04.25Nx, 04.70Bw I. INTRODUCTIONComputational methods are essential to provide the waveform from the coalescence of black holes, which is expected to produce a strong signal for the gravitational wave observatories being developed. The importance of the binary black hole problem to the success of LIGO and LISA has led to major computational efforts, most notably the Binary Black Hole Grand Challenge. Although this Grand Challenge had intermediate successes [1,2,3], scientifically useful waveforms were not delivered. At present, this remains a problem beyond the reach of any existing code.A recent study [4] of large scale scientific code projects at the Livermore, Los Alamos and Sandia National Laboratories, funded under the U.S. Department of Energy's Accelerated Strategic Computing Initiative (ASCI), identified three necessary elements for success: verification, validation and quality management. In the absence of any of those three requirements, the report concluded that the results would have little scientific impact because of the impossibility to judge code reliability. Although the ASCI projects involved highly experienced and qualified teams at laboratories with ample resources, only one third of the projects succeeded as planned, another third succeeded later than planned and the remaining were eventually abandoned. The failed projects had overly ambitious schedules and goals and lacked a conservative methodology that minimized risk. It was expected that the failure rate would have been much higher ...