4.1 Computational BoundariesCharacteristic Evolution and Matching3.6 Characteristic Treatment of Binary

4 Cauchy-Characteristic Matching 

Characteristic evolution has many advantages over Cauchy evolution. Its one disadvantage is the existence of either caustics where neighboring characteristics focus or, a milder version consisting of 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. It does not seem possible for a single characteristic coordinate system to cover the entire exterior region of a binary black hole spacetime without developing more complicated caustics or crossovers. This limits the waveform determined by a purely characteristic evolution to the post merger period.

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 [25Jump To The Next Citation Point In The Article]. 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:

  1. accurate waveform and polarization properties at infinity,
  2. computational efficiency for radiation problems in terms of both the grid domain and the computational algorithm,
  3. 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
  4. 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:

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 BoundariesCharacteristic Evolution and Matching3.6 Characteristic Treatment of Binary

image Characteristic Evolution and Matching
Jeffrey Winicour
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