The most important application of CCM is anticipated to be the waveform and momentum recoil in the binary black-hole inspiral and merger. The 3D Cauchy codes being applied to simulate this problem employ a single Cartesian coordinate patch. In principle, the application of CCM to this problem might seem routine, tantamount to translating into finite-difference form the textbook construction of an atlas consisting of overlapping coordinate patches. In practice, it is a complicated project. The computational strategy has been outlined in [52]. The underlying algorithm consists of the following main submodules:

- The boundary module, which sets the grid structures. This defines masks identifying which points in the Cauchy grid are to be evolved by the Cauchy module and which points are to be interpolated from the characteristic grid, and vice versa. The reference structures for constructing the mask is the inner characteristic boundary, which in the Cartesian Cauchy coordinates is the “ spherical” extraction worldtube , and the outer Cauchy boundary , where the Cauchy boundary data is injected. The choice of lapse and shift for the Cauchy evolution governs the dynamical and geometrical properties of these worldtubes.
- The extraction module whose input is Cauchy grid data in the neighborhood of the extraction worldtube at and whose output is the inner boundary data for the exterior characteristic evolution. This module numerically implements the transformation from Cartesian {3 + 1} coordinates to spherical null coordinates. The algorithm makes no perturbative assumptions and is based upon interpolations of the Cauchy data to a set of prescribed grid points near . The metric information is then used to solve for the null geodesics normal to the slices of the extraction worldtube. This provides the Jacobian for the transformation to null coordinates in the neighborhood of the worldtube. The characteristic evolution module is then used to propagate the data from the worldtube to null infinity, where the waveform is calculated.
- The injection module, which completes the interface by using the exterior characteristic evolution to inject the outer boundary data for the Cauchy evolution at . This is the inverse of the extraction procedure but must be implemented with to allow for overlap between the Cauchy and characteristic domains. The overlap region can be constructed either to have a fixed physical size or to shrink to zero in the continuum limit. In the latter case, the inverse Jacobian describing the transformation from null to Cauchy coordinates can be obtained to prescribed accuracy in terms of an affine parameter expansion along the null geodesics emanating from the worldtube. The numerical stability of this element of the scheme is not guaranteed.

The above strategy provides a model of how Cauchy and characteristic codes can be pieced together as modules to form a global evolution code.

The full advantage of CCM lies in the numerical treatment of nonlinear systems, where its error converges to zero in the continuum limit for any size outer boundary and extraction radius [45, 46, 89]. For high accuracy, CCM is also very efficient. For small target error , it has been shown on the assumption of unigrid codes that the relative amount of computation required for CCM () compared to that required for a pure Cauchy calculation () goes to zero, as [56, 52]. An important factor here is the use of a compactified characteristic evolution, so that the whole spacetime is represented on a finite grid. From a numerical point of view this means that the only error made in a calculation of the radiation waveform at infinity is the controlled error due to the finite discretization.

The accuracy of a Cauchy algorithm, which uses an ABC, requires a large grid domain in order to avoid error from nonlinear effects in its exterior. Improved numerical techniques, such as the design of Cauchy grids whose resolution decreases with radius, has improved the efficiency of this approach. Nevertheless, the computational demands of CCM are small since the interface problem involves one less dimension than the evolution problem and characteristic evolution algorithms are more efficient than Cauchy algorithms. CCM also offers the possibility of using a small matching radius, consistent with the requirement that it lie in the region exterior to any caustics. This is advantageous in simulations of stellar collapse, in which the star extends over the entire computational grid, although it is then necessary to include the matter in the characteristic treatment.

At present, the computational strategy of CCM is mainly the tool of numerical relativists, who are used to dealing with dynamical coordinate systems. The first discussion of its potential was given in [45] and its feasibility has been more fully explored in [89, 90, 102, 49, 287]. Recent work has been stimulated by the requirements of the binary black-hole problem, where CCM is one of the strategies to provide boundary conditions and determine the radiation waveform. However, it also has inherent advantages in dealing with other hyperbolic systems in computational physics, particularly nonlinear three-dimensional problems. A detailed study of the stability and accuracy of CCM for linear and nonlinear wave equations has been presented in [50], illustrating its potential for a wide range of problems.

Living Rev. Relativity 15, (2012), 2
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