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  to avoid coordinate singularites. A thoroughly tested and robust 3D characteristic code is now in place , 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 . The underlying geometrical algorithm consists of the following main submodules:
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 . 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 , illustrating its potential for a wide range of problems.
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