CCM is a way to avoid such limitations by combining the strong points of characteristic and Cauchy evolution into a global evolution [33]. 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 state 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 achieved in fully nonlinear three-dimensional general relativity. The early attempts to implement CCM in general relativity involved the Arnowitt-Deser-Misner (ADM) [12] formulation for the Cauchy evolution. The difficulties were later traced to a pathology in the way boundary conditions have traditionally been applied in ADM codes. Instabilities introduced at boundaries have emerged as a major problem common to all ADM code development. A linearized study [206, 207] of ADM evolution-boundary algorithms with prescribed values of lapse and shift shows the following:

- On analytic grounds, ADM boundary algorithms which supply values for all components of the metric (or extrinsic curvature) are inconsistent.
- A consistent boundary algorithm only allows free specification of the transverse-traceless components of the metric with respect to the boundary.
- Using such a boundary algorithm, linearized ADM evolution can be carried out in a bounded domain for thousands of crossing times with no sign of exponential growing instability.

The evolution satisfied the original criterion for robust stability [207]: that there be no exponential growth when the initial Cauchy data and free boundary data are prescribed as random numbers (in the linearized regime). These results gave some initial optimism that CCM might be possible with an ADM code if the boundary condition was properly treated. However, it was subsequently shown that ADM is only weakly hyperbolic so that in the linear regime there are instabilities which grow as a power law in time. In the nonlinear regime, it is symptomatic of weakly hyperbolic systems that such secular instabilities become exponential. This has led to a refined criterion for robust stability as a standardized test [5].

CCM cannot work unless the Cauchy and characteristic codes have robustly stable boundaries. This is necessarily so because interpolations continually introduce short wavelength noise into the neighborhood of the boundary. It was demonstrated some time ago that the PITT characteristic code has a robustly stable boundary (see Section 3.5.3), but robustness of the Cauchy boundary has only recently been studied.

4.1 Computational boundaries

4.2 The computational matching strategy

4.3 Perturbative matching schemes

4.4 Cauchy-characteristic matching for 1D gravitational systems

4.4.1 Cylindrical matching

4.4.2 Spherical matching

4.4.3 Excising 1D black holes

4.5 Axisymmetric Cauchy-characteristic matching

4.6 Cauchy-characteristic matching for 3D scalar waves

4.7 Stable implementation of 3D linearized Cauchy-characteristic matching

4.8 The binary black hole inner boundary

http://www.livingreviews.org/lrr-2005-10 |
© Max Planck Society and the author(s)
Problems/comments to |