6 Cauchy-Characteristic Extraction of Waveforms

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 Ψ 4, 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 Ψ4 (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 [199Jump To The Next Citation Point] 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 [51Jump To The Next Citation Point], 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 Ψ4 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 [237Jump To The Next Citation Point].

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 Φ, the accuracies of the circular-stereographic and cubed-sphere methods were compared [13Jump To The Next Citation Point]. For equivalent computational expense, the cubed-sphere error in the scalar field ℰ (Φ ) was ≈ 13 the circular-stereographic error but the advantage was smaller for the higher ∂-derivatives (angular derivatives) required in gravitational waveform extraction. The cubed-sphere error ℰ (∂¯∂2Φ ) was ≈ 4 5 the stereographic error. However, the cubed-sphere method has not yet been developed for extraction of gravitational waveforms at + ℐ.

 6.1 Waveforms at null infinity
 6.2 Application of CCE to binary black hole inspirals
 6.3 Application of CCE to stellar collapse
 6.4 LIGO accuracy standards
 6.5 A community CCE tool
 6.6 Initial characteristic data for CCE

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