The matching interface was streamlined by introducing a pseudospectral decomposition of the Cauchy
metric in the neighborhood of the extraction worldtube. This provides economical storage of the boundary
data for the characteristic code so that the waveform at can be obtained in post-processing with a
small computational burden compared to the Cauchy evolution. The new version incorporates stereographic
grids with circular patch boundaries [13], which eliminates the large error from the corners of the square
patches used previously. The finite-difference accuracy of the angular derivatives was increased to fourth
order. Bugs were eliminated that had been introduced in the process of parallelizing the code
using the Cactus framework [292]. In addition, the worldtube module, which supplies the inner
boundary data for the characteristic evolution, was revamped so that it provides a consistent,
second-order–accurate start-up algorithm for numerically-generated Cauchy data. The prior module
required differentiable Cauchy data, as provided by analytic testbeds, to be consistent with
convergence.
These changes led to clean second-order convergence of all evolved quantities at finite locations. Because
some of the hypersurface equations become degenerate at , certain asymptotic quantities, in particular
the Bondi news function, are only first-order accurate. However, the clean first-order convergence allows the
application of Richardson extrapolation, based upon three characteristic grid sizes, to extract waveforms
with third-order accuracy.
The error norm for the extracted news function, as defined in Equation (85
), has been
measured for the simulation of the inspiral of equal mass, non-spinning black holes obtained via a BSSN
simulation [16
]. The advanced LIGO criterion for detection (87
), was satisfied for
(which
corresponds to less than a 10% signal loss) and for values of
throughout the entire binary mass range.
The criterion (88
) for measurement is more stringent. For the expected lower bound of the
calibration factor
, for
(corresponding to the most demanding small
mass limit) and for the most optimistic advanced LIGO signal-to-noise ratio
, the
requirement for measurement is
. This measurement criterion was satisfied
throughout the entire binary mass range by the numerical truncation error
in the CCE
waveform.
These detection and measurement criteria were satisfied for a range of extraction worldtubes extending
from to
. The
error norm decreased with larger extraction radius, as
expected since the error introduced by characteristic evolution depends upon the size of the integration
region between the extraction worldtube and
. However, the modeling error corresponding to the
difference in waveforms obtained with extraction at
as compared to
only
satisfied the measurement criterion for signal-to-noise ratios
(which would still cover the most
likely advanced LIGO events). This modeling error results from the different initial data, which
correspond to different extraction radii. This error would be smaller for longer simulations with a
higher number of orbits. The results suggest that the choice of extraction radius should be
balanced between a sufficiently large radius to reduce initialization effects and a sufficiently small
radius where the Cauchy grid is more highly refined and outer boundary effects are better
isolated.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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