Characteristic evolution has many advantages over Cauchy evolution. Its main disadvantage is the existence of either a caustic, where neighboring characteristics focus, or a milder version consisting of a crossover between two distinct characteristics. The vertex of a light cone is a highly symmetric caustic, which already strongly limits the time step for characteristic evolution because of the CFL condition (22). It does not appear possible for a single characteristic coordinate system to cover the entire exterior region of a binary black-hole spacetime without developing very complicated caustics and crossovers. This limits the waveform determined by a purely characteristic evolution to the post merger period.

CCM is a way to avoid this limitation by combining the strong points of characteristic and Cauchy evolution into a global evolution [46]. One of the prime goals of computational relativity is the simulation of the inspiral and merger of binary black holes. Given the appropriate data on a worldtube surrounding a binary system, 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 event horizon.

These advantages have been realized in model tests (see Sections 5.5 – 5.8), but CCM has not yet been achieved in fully nonlinear three-dimensional general relativity. The early attempts to implement CCM in general relativity employed the Arnowitt–Deser–Misner (ADM) [12] formulation, with explicit lapse and shift, for the Cauchy evolution. A major problem in this application has since been identified with the weakly hyperbolic nature of this system. Even at the analytic level of the Cauchy problem there are secularly growing modes with arbitrarily fast rates, i.e., the Cauchy problem is ill-posed. Such power-law instabilities of the Cauchy problem can be converted to exponentially growing instabilities by the introduction of lower order or nonlinear terms. See [76] for discussions relevant to the stability of the ADM formulation.

Such behavior can also be made worse by the imposition of boundary conditions. Linearized studies [284, 285, 18] of ADM evolution-boundary algorithms with prescribed values of lapse and shift have shown the following:

- On analytic grounds, those ADM boundary algorithms that supply values for all components of the metric (or extrinsic curvature) are inconsistent.
- A consistent boundary algorithm 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 without sign of exponential growth, even though there are the secularly growing modes whose rates increase with resolution.

Such results contributed to the early belief that long term evolutions might be possible by means of ADM evolution. The linearized tests satisfied the original criterion for robust stability, i.e., that there be no exponential growth when the initial Cauchy data and free boundary data are prescribed as random numbers (in the linearized regime) [285]. However, it was subsequently shown that the weakly hyperbolic nature of ADM led to uncontrolled power law instabilities. In the nonlinear regime, it is symptomatic of weakly hyperbolic systems that such instabilities become exponential. This has led to refined criteria for robust stability as a standardized test [18].

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 has been demonstrated that the PITT characteristic code has a robustly-stable boundary (see Section 4.2.5), but robustness of the Cauchy boundary has only recently been studied.

5.1 Computational boundaries

5.2 The computational matching strategy

5.3 The outer Cauchy boundary in numerical relativity

5.4 Perturbative matching schemes

5.5 Cauchy-characteristic matching for 1D gravitational systems

5.5.1 Cylindrical matching

5.5.2 Spherical matching

5.5.3 Excising 1D black holes

5.6 Axisymmetric Cauchy-characteristic matching

5.7 Cauchy-characteristic matching for 3D scalar waves

5.8 Stable 3D linearized Cauchy-characteristic matching

5.9 The binary black-hole inner boundary

5.2 The computational matching strategy

5.3 The outer Cauchy boundary in numerical relativity

5.4 Perturbative matching schemes

5.5 Cauchy-characteristic matching for 1D gravitational systems

5.5.1 Cylindrical matching

5.5.2 Spherical matching

5.5.3 Excising 1D black holes

5.6 Axisymmetric Cauchy-characteristic matching

5.7 Cauchy-characteristic matching for 3D scalar waves

5.8 Stable 3D linearized Cauchy-characteristic matching

5.9 The binary black-hole inner boundary

Living Rev. Relativity 15, (2012), 2
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