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 [103]. 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 [52Jump 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 for any size outer boundary and extraction radius [45Jump To The Next Citation Point, 46, 89Jump To The Next Citation Point]. For high accuracy, CCM is also very efficient. For small target error 𝜀, it has been shown on the assumption of unigrid codes 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 [56Jump To The Next Citation Point, 52Jump 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.

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 [89Jump To The Next Citation Point, 90Jump To The Next Citation Point, 102Jump To The Next Citation Point, 49Jump To The Next Citation Point, 287Jump 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 three-dimensional problems. A detailed study of the stability and accuracy of CCM for linear and nonlinear wave equations has been presented in [50Jump To The Next Citation Point], illustrating its potential for a wide range of problems.

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