When an artificial finite outer boundary is introduced there are two broad sources of error:
CCM addresses both of these items. Cauchy-characteristic extraction (CCE), which is one of the pieces
of the CCM strategy, offers a means to avoid the second source of error introduced by extraction at a finite
worldtube. In current codes used to simulate black holes, the waveform is extracted at an interior
worldtube, which must be sufficiently far inside the outer boundary in order to isolate it from errors
introduced by the boundary condition. At this inner worldtube, the waveform is extracted by a perturbative
scheme based upon the introduction of a background Schwarzschild spacetime. This has been carried out
using the Regge–Wheeler–Zerilli [240, 309] treatment of the perturbed metric, as reviewed in [214], and
also by calculating the Newman–Penrose Weyl component , as first done for the binary black-hole
problem in [19, 235, 78, 20]. In these approaches, errors arise from the finite size of the extraction
worldtube, from nonlinearities and from gauge ambiguities involved in the arbitrary introduction of
a background metric. The gauge ambiguities might seem less severe in the case of
(vs
metric) extraction, but there are still delicate problems associated with the choices of a preferred
null tetrad and preferred worldlines along which to measure the waveform (see [199
] for an
analysis).
CCE offers a means to avoid this error introduced by extraction at a finite worldtube. In CCE, the inner
worldtube data supplied by the Cauchy evolution is used as boundary data for a characteristic evolution to
future null infinity, where the waveform can be unambiguously computed in terms of the Bondi news
function. By itself, CCE does not use the characteristic evolution to inject outer boundary data for the
Cauchy evolution, which can be a source of instability in full CCM. A wide number of highly nonlinear tests
involving black holes [56, 53, 311, 312] have shown that early versions of CCE were a stable
procedure, which provided the gravitational waveform up to numerical error that is second-order
convergent when the worldtube data is prescribed in analytic form. Nevertheless, in nonlinear
applications requiring numerical worldtube data and high resolution, such as the inspiral of
matter into a black hole [51], the numerical error was a troublesome factor in computing the
waveform. The CCE modules were first developed in a past period when stability was the dominant
issue and second-order accuracy was considered sufficient. Only recently have they begun to be
updated to include the more accurate techniques now standard in Cauchy codes. There are two
distinct ways, geometric and numerical, that the accuracy of CCE might be improved. In the
geometrical category, one option is to compute
instead of the news function as the primary
description of the waveform. In the numerical category, some standard methods for improving
accuracy, such as higher-order finite difference approximations, are straightforward to implement
whereas others, such as adaptive mesh refinement, have only been tackled for 1D characteristic
codes [237
].
A major source of numerical error in characteristic evolution arises from the intergrid interpolations
arising from the multiple patches necessary to coordinatize the spherical cross-sections of the outgoing null
hypersurfaces. More accurate methods, have now been developed to reduce this interpolation error, as
discussed in Section 4.1. In particular, the cubed-sphere method and the stereographic method with
circular patch boundaries have both shown improvement over the original use of square stereographic
patches. In a test problem involving a scalar wave , the accuracies of the circular-stereographic and
cubed-sphere methods were compared [13
]. For equivalent computational expense, the cubed-sphere error
in the scalar field
was
the circular-stereographic error but the advantage was
smaller for the higher
-derivatives (angular derivatives) required in gravitational waveform
extraction. The cubed-sphere error
was
the stereographic error. However, the
cubed-sphere method has not yet been developed for extraction of gravitational waveforms at
.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
![]() This work is licensed under a Creative Commons License. E-mail us: |