### 4.5 Axisymmetric Cauchy-characteristic matching

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 [74, 75]. Similar to the
Pittsburgh 3D strategy (see Section 4.2), 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 coordinate singularity in the Cauchy evolution. The
characteristic code (see Section 3.3.3) is based upon a compactified Bondi-Sachs formalism. The use of
a “radial” Cauchy gauge, in which the Cauchy coordinate measures the surface area of
spheres, simplifies the relation to the Bondi-Sachs 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.
Preliminary results in the development of the Southampton CCM code are described by Pollney in his
thesis [172]. The Cauchy code is based upon the axisymmetric ADM code of Stark and Piran [200] 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
boost-rotation symmetric solutions [30], with more thorough tests of stability and accuracy still to be
carried out.