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4.2 The computational matching strategy

In its simplest form, CCM evolves a mixed spacelike-null initial value problem in which Cauchy data is given in a spacelike region bounded by a spherical boundary S and characteristic data is given on a null hypersurface emanating from S. The general idea is not entirely new. An early mathematical investigation combining space-like and characteristic hypersurfaces appears in the work of Duff [77]. The three chief ingredients for computational implementation are: (i) a Cauchy evolution module, (ii) a characteristic evolution module and, (iii) a module for matching the Cauchy and characteristic regions across their interface. The interface is the timelike worldtube which is traced out by the flow of S along the worldlines of the Cauchy evolution, as determined by the choice of lapse and shift. Matching provides the exchange of data across the worldtube to allow evolution without any further boundary conditions, as would be necessary in either a purely Cauchy or purely characteristic evolution. Other versions of CCM involve a finite overlap between the characteristic and Cauchy regions.

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 [31Jump To The Next Citation Point], 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 [39Jump To The Next Citation Point]. 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 [32Jump To The Next Citation Point3366Jump To The Next Citation Point]. For high accuracy, CCM is also by far the most efficient method. For small target error e, it has been shown that the relative amount of computation required for CCM (ACCM) compared to that required for a pure Cauchy calculation (AC) goes to zero, ACCM/AC --> O as e --> O [42Jump To The Next Citation Point39Jump To The Next Citation Point]. 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 [66Jump To The Next Citation Point67Jump To The Next Citation Point76Jump To The Next Citation Point36Jump To The Next Citation Point208Jump To The Next Citation Point] 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 [37Jump To The Next Citation Point], illustrating its potential for a wide range of problems.

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