From a cursory view, the application of CCM to this problem might seem routine, tantamount to translating into finite difference form the textbook construction of an atlas consisting of overlapping coordinate patches. In practice, it is an enormous project, which would not have been possible to undertake without the impetus and support of the Binary Black Hole Grand Challenge.

A CCM module has been constructed and interfaced with Cauchy and characteristic evolution modules. It provides a model of how two of the best current codes to treat gravitation, the Grand Challenge ADM and characteristic codes, can be pieced together as modules to form a single global code. The documentation of the underlying geometrical algorithm is given in Ref. [91]. The main submodules of the CCM module are:

- The
**outer boundary module**which sets the grid structures. This defines masks identifying which points in the Cauchy grid are to be evolved by the Cauchy module and which points are to be interpolated from the characteristic grid; and vice versa. The reference base for constructing the mask is the matching worldtube, which in Cartesian coordinates is the ``Euclidean'' sphere . The choice of lapse and shift for the Cauchy evolution govern the actual dynamical and geometrical properties of the matching worldtube. - The
**extraction module**whose input is Cauchy grid data in the neighborhood of the worldtube and whose output is the inner boundary data for the exterior characteristic evolution. This module numerically implements the transformation from Cartesian ``3+1'' coordinates to spherical null coordinates. The algorithm makes no perturbative assumptions and is based upon interpolations of the Cauchy data to a set of prescribed points on the worldtube. The metric information is then used to solve for the null geodesics normal to the slices of the worldtube. This provides the Jacobian for the transformation to null coordinates in the neighborhood of the worldtube. The characteristic evolution module is then used to propagate the data from the worldtube to null infinity, where the waveform is calculated. - The
**injection module**which completes the interface by using the exterior characteristic evolution to supply the outer boundary condition for Cauchy evolution. This is the inverse of the extraction procedure but must be implemented outside the worldtube to allow for the necessary overlap between Cauchy and characteristic domains. (Without overlap, the domain of dependence of the initial value problem would be empty.) The overlap region is constructed so that it shrinks to zero in the continuum limit. As a result, the inverse Jacobian can be obtained to a prescribed accuracy in terms of an affine parameter expansion of the null geodesics about the worldtube.

The CCM module has been calibrated to give a second order accurate interface between Cauchy and characteristic evolution modules. When its long term stability has been established, it will provide an accurate outer boundary condition for an interior Cauchy evolution by joining it to an exterior characteristic evolution which extracts the waveform at infinity.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-1998-5
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