4.7 3D Cauchy-Characteristic Matching

The most important application of CCM is anticipated to be the binary black hole problem. The 3D Cauchy codes now being developed to solve this problem employ a single Cartesian coordinate patch [141]. A thoroughly tested and robust 3D characteristic code is now in place [30], ready to match to the boundaries of this Cauchy patch. Development of a stable implementation of CCM represents the major step necessary to provide a global code for the binary problem.

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.

A CCM module has been constructed and interfaced with Cauchy and characteristic evolution modules. It provides a model of how Cauchy 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. [28]. 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 governs the 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 overlap between Cauchy and characteristic domains. 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 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-2001-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de