The most important application of CCM is anticipated to be the binary black hole problem. The 3D Cauchy codes being developed to solve this problem employ a single Cartesian coordinate patch, a stategy adopted in [68] to avoid coordinate singularites. A thoroughly tested and robust 3D characteristic code is now in place [31], ready to match to the boundary of this Cauchy patch. Development of a stable implementation of CCM represents the major step necessary to provide a global evolution code for the binary black hole problem.

From a cursory view, 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 an enormous project. The computational strategy has been outlined in [39]. The underlying geometrical 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 Cartesian coordinates is the “Euclidean” spherical worldtube , and the outer Cauchy boundary. The choice of lapse and shift for the Cauchy evolution governs the dynamical and geometrical properties of the matching worldtube.
- The extraction module whose input is Cauchy grid data in the neighborhood of the worldtube 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 on the worldtube. The metric information is then used to solve for the null geodesics normal to the slices of the 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 supply the outer boundary data for the Cauchy evolution. This is the inverse of the extraction procedure but must be implemented outside the worldtube to allow for overlap between 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. But the numerical stability 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 of infinite grid resolution [32, 33, 66]. For high accuracy, CCM is also by far the most efficient method. For small target error , it has been shown that the relative amount of computation required for CCM () compared to that required for a pure Cauchy calculation () goes to zero, as [42, 39]. 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. 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. The computational demands of CCM are small because the interface problem involves one less dimension than the evolution problem. Because characteristic evolution algorithms are more efficient than Cauchy algorithms, the efficiency can be further enhanced by making the matching radius as small as possible consistent with the avoidance of caustics.

At present, the computational strategy of CCM is exclusively the tool of general relativists who are used to dealing with novel coordinate systems. A discussion of its potential is given in [32]. Only recently [66, 67, 76, 36, 208] has its practicability been carefully explored. Research on this topic has been stimulated by the requirements of the Binary Black Hole Grand Challenge Alliance, where CCM was one of the strategies pursued to provide boundary conditions and determine the radiation waveform. But I anticipate that its use will eventually spread throughout computational physics because of its inherent advantages in dealing with hyperbolic systems, particularly in three-dimensional problems where efficiency is desired. A detailed study of the stability and accuracy of CCM for linear and nonlinear wave equations has been presented in [37], illustrating its potential for a wide range of problems.

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