Cauchy-characteristic matching (CCM) is a way to avoid such limitations by combining the strong points of characteristic and Cauchy evolution into a global evolution [25]. One of the prime goals of computational relativity is the simulation of the inspiral and merger of binary black holes. Given the appropriate worldtube data for a binary system in its interior, characteristic evolution can supply the exterior spacetime and the radiated waveform. But determination of the worldtube data for a binary requires an interior Cauchy evolution. CCM is designed to solve such global problems. The potential advantages of CCM over traditional boundary conditions are:

- accurate waveform and polarization properties at infinity,
- computational efficiency for radiation problems in terms of both the grid domain and the computational algorithm,
- elimination of an artificial outer boundary condition on the Cauchy problem, which eliminates contamination from back reflection and clarifies the global initial value problem, and
- a global picture of the spacetime exterior to the horizon.

These advantages have been realized in model tests but CCM has not yet been successful in either axisymmetric or fully three-dimensional general relativity. This difficulty may possibly arise from a pathology in the way boundary conditions have traditionally been applied in the Arnowitt-Deser-Misner (ADM) [10] formulation of the Einstein equations which, at present, is the only formulation for which CCM has been attempted.

Instabilities or inaccuracies introduced at boundaries have emerged as a major problem common to all ADM code development and have led to pessimism that such codes might be inherently unstable because of the lack of manifest hyperbolicity in the underlying equations. In order to shed light on this issue, B. Szilágyi [137, 138], as part of his thesis research, carried out a study of ADM evolution-boundary algorithms in the simple environment of linearized gravity, where nonlinear sources of physical or numerical instability are absent and computing time is reduced by a factor of five by use of a linearized code. The two main results, for prescribed values of lapse and shift, were:

- On analytic grounds, ADM boundary algorithms which supply values for all components of the metric (or extrinsic curvature) are inconsistent.
- Using a consistent boundary algorithm, which only allows free specification of the transverse-traceless components of the metric with respect to the boundary, an unconstrained, linearized ADM evolution can be carried out in a bounded domain for thousands of crossing times with robust stability.

The criteria for robust stability is that the initial Cauchy data and free boundary data be prescribed as random numbers. It is the most severe test of stability yet carried out in the Cauchy evolution of general relativity. Similar robust stability tests were previously successfully carried out for the PITT characteristic code.

CCM cannot work unless the Cauchy code, as well as the characteristic code, has a robustly stable boundary. This is necessarily so because interpolations continually introduce short wavelength noise into the neighborhood of the boundary. Robustness of the Cauchy boundary is a necessary (although not a sufficient) condition for the successful implementation of CCM. The robustly stable ADM evolution-boundary algorithm differs from previous approaches and offers fresh hope for the success of CCM in general relativity.

- 4.1 Computational Boundaries
- 4.2 Perturbative Cauchy-Characteristic Matching
- 4.3 Analytic-Numerical Matching for Waves
- 4.4 Numerical Matching for 1D Gravitational Systems
- 4.5 Axisymmetric Cauchy-Characteristic Matching
- 4.6 Cauchy-Characteristic Matching for 3D Scalar Waves
- 4.7 3D Cauchy-Characteristic Matching
- 4.8 The Binary Black Hole Inner Boundary

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
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