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 (11). 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 [39]. 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 3-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 major problem was later traced to a pathology in the way boundary conditions have traditionally been applied in ADM codes. Exponentially growing instabilities introduced at boundaries have emerged as a major problem common to all ADM code development.

Linearized studies [234, 235, 15] of ADM evolution-boundary algorithms with prescribed values of lapse and shift show the following:

- On analytic grounds, those ADM boundary algorithms which 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 an exponential growing instability but with error that grows secularly in time.

The linearized evolution satisfied the original criterion for robust stability that there be no exponential growth when the initial Cauchy data and free boundary data are prescribed as random numbers (in the linearized regime) [235] . 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 refined criteria for robust stability as a standardized test [15].

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 4.2.4), 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

http://www.livingreviews.org/lrr-2009-3 |
This work is licensed under a Creative Commons License. Problems/comments to |