Well-posedness of the IBVP, in addition to constraint preservation, is a necessary condition for computational success. This is particularly cogent for dealing with waveform extraction in the simulation of black holes by BSSN or harmonic formulations. There is no well-posed outer boundary theory for the BSSN formulation and the strategy is to place the boundary out far enough so that it does no harm. The harmonic formulation has a simpler mathematical structure as a system of coupled quasilinear wave equations which is more amenable to an analytic treatment.

Standard harmonic coordinates satisfy the covariant wave equation

(This can easily be generalized to include gauge forcing [97], whereby . For simplicity of discussion, I will set , although gauge forcing is an essential tool in simulating black holes [195].)When , Einstein’s equations reduce to the 10 quasilinear wave equations

where does not enter the principle part and vanishes in the linearized approximation. Straightforward techniques can be applied to formulate a well-posed IBVP for the system (45). The catch is that Einstein’s equations are not necessarily satisfied unless the constraints are also satisfied.In the harmonic formalism, the constraints can be reduced to the harmonic coordinate conditions (44). For the resulting IBVP to be constraint preserving, these harmonic conditions must be built into the boundary condition. Numerous early attempts to accomplish this failed because (44) contains derivatives tangent to the boundary, which do not fit into the standard methods for obtaining the necessary energy estimates. The use of pseudo-differential techniques developed for similar problems in elasticity theory has led to the first well-posed formulation for the general IBVP for the harmonic Einstein equations [162]. Subsequently, these results were also obtained using standard energy estimates by means of a novel, non-conventional choice of the energy for the harmonic system [160]. Furthermore, the allowed boundary conditions include those of the Sommerfeld type which are nonreflecting in the sense that the boundary data for falls off as as the boundary radius [161]

A Cauchy evolution code, the Abigel code, has been based upon a discretized of this well-posed harmonic IBVP [236]. The code was tested to be stable, convergent and constraint preserving in the nonlinear regime [16]. A linearized version of the Abigel code has been used to successfully carry out CCM (see Section 5.8).

In the present harmonic codes used to simulate the binary black holes, the best that can be done is to impose a constraint preserving boundary condition for which homogeneous boundary data, i.e. zero boundary values, is a good approximation. One proposal of this type [98] is a boundary condition that requires the Newman–Penrose [179] Weyl tensor component to vanish. In the limit that the outer boundary goes to infinity this outer boundary condition becomes exact. In the present state of the art of black hole simulations, this approach comes closest to a satisfactory treatment of the outer boundary [205].

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