

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 [52]. A thoroughly tested and robust 3D characteristic code is now
in place [60
], ready to match to the boundaries of this Cauchy patch.
Development of a stable implementation of CCM represents the last
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, 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.


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Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-1998-5
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to
livrev@aei-potsdam.mpg.de
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