The Southampton CCM project is being carried out for spacetimes
with (twisting) axial symmetry. The formal basis for the matching
scheme was developed by d'Inverno and Vickers [52,
53]. Similar to the Pittsburgh 3D strategy (see Sec.
4.7), matching is based upon an extraction module, which supplies
boundary data for the exterior characteristic evolution, and an
injection module, which supplies boundary data for the interior
Cauchy evolution. However, their use of spherical coordinates for
the Cauchy evolution (as opposed to Cartesian coordinates in the
3D strategy) allows use of a matching worldtube
which lies simultaneously on Cauchy and characteristic
gridpoints. This tremendously simplifies the necessary
interpolations between the Cauchy and characteristic evolutions,
at the expense of dealing with the
r
=0 coordinate singularity in the Cauchy evolution. The
characteristic code (see Sec.
3.3.2) is based upon a compactified BondiSachs formalism. The use of
a ``radial'' Cauchy gauge, in which the Cauchy coordinate
r
measures the surface area of spheres, simplifies the relation to
the BondiSachs coordinates. In the numerical scheme, the metric
and its derivatives are passed between the Cauchy and
characteristic evolutions exactly at
, thus eliminating the need of a matching interface encompassing
a few grid zones, as in the 3D Pittsburgh scheme. This avoids a
great deal of interpolation error and computational complexity.
The implementation of the Southampton CCM algorithm into a
running code is nearing completion, with preliminary results
described by D. Pollney in his thesis [115]. The Cauchy code is based upon the axisymmetric ADM code of
Stark and Piran [132] and reproduces their vacuum results for a short time period,
after which an instability at the origin becomes manifest. The
characteristic code has been tested to reproduce accurately the
Schwarzschild and boostrotation symmetric solutions [23], with more thorough tests of stability and accuracy still being
carried out. Much progress has been made but much work remains to
make the code useful for scientific application.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr20013
© MaxPlanckGesellschaft. ISSN 14338351
Problems/Comments to
livrev@aeipotsdam.mpg.de
