5.2 The computational matching strategy

CCM evolves a mixed spacelike-null initial value problem in which Cauchy data is given in a spacelike hypersurface 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 spacelike and characteristic hypersurfaces appears in the work of Duff [88]. 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. In the simplest scenario, 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 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 [45Jump To The Next Citation Point]. The underlying 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 [38Jump To The Next Citation Point3976Jump To The Next Citation Point]. For high accuracy, CCM is also the most efficient method. For small target error 𝜀, 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 𝜀 → O [49Jump To The Next Citation Point45Jump 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 mainly the tool of numerical relativists, who are used to dealing with dynamical coordinate systems. The first discussion of its potential was given in [38] and its feasibility has been more fully explored in [76Jump To The Next Citation Point77Jump To The Next Citation Point87Jump To The Next Citation Point42Jump To The Next Citation Point236Jump To The Next Citation Point]. 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 3-dimensional problems. A detailed study of the stability and accuracy of CCM for linear and nonlinear wave equations has been presented in [43Jump To The Next Citation Point], illustrating its potential for a wide range of problems.

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