APPARENT HORIZONS AND INNER BOUNDARIES<br>
summarized by A. Abrahams and C. Evans<br>


    SUMMARY:

The basic idea is that the apparent horizon will be located, at least
approximately, each timestep and that the interior of the horizon will 
be excised from the computational domain.  This will entail some sort 
of one-sided difference scheme around the horizon.  An important element 
of the scheme is the choice of a shift vector that allows the black hole 
orbital dynamics to be followed while the spatial coordinates remain 
regular and that provides a good discretization of the exterior region.  
For example, the coordinate areas of the holes could be roughly conserved 
in time while the holes advect through the grid.<br>


    SPECIFICS:

Shift Vector Choice:

The shift vector or spatial coordinate choice seems to be of fundamental
importance.  Intuitively it seems that the shift vector should be a 
superposition of two parts.  The first part is a vector field that is 
outward pointing on each horizon with magnitude close to the coordinate 
speed of light and falling off towards zero at large distances.  This 
component serves approximately to hold the coordinate area of the holes 
constant.  Superposed on this is a vector field with sinusoidal angular 
dependence (locally uniform vector field) at the horizon, which also 
decreases with distance.  This component of the shift vector allows grid 
points to flux into the horizon on one side and reemerge from the other, 
while preserving the constant coordinate area condition.  With the proper
global dependence to the shift, the holes would orbit in the mesh.  It 
seems likely that this sort of shift will be most naturally formulated as 
an elliptic boundary value problem, with the shift vector the
gradient of the scalar solution.  

The shift should be spatially and temporally smooth so that second-order 
convergence can be achieved within the adaptive mesh framework.  
These considerations may motivate a multipole expansion solution
to the elliptic equation similar in spirit to that discussed below 
for the apparent horizon equation.

Apparent Horizon Finder:

Two means have been developed within the alliance to locate apparent 
horizons.  One method, developed at Texas, is based on discretizing a 
two-surface, embedding it in the three-space, and solving the apparent 
horizon equation for the shape of this surface by a Newton-Raphson method.  
The second method, under development at NCSA and Cornell, uses a 
multipole-moment (spectral) decomposition of the apparent horizon and 
solves for the coefficients of the multipole expansion.  The techniques 
appear ready to be tried in more elaborate applications.

It seems likely that we will need to locate and excise the horizon on the 
finest mesh on each of the finest time-steps.  Indeed, the black holes 
may not be ``visible'' on the coarsest mesh.  It may not be necessary to 
have a highly accurate location of the apparent horizon, as long as the 
approximate surface is close but strictly interior to it.  More important 
may be to have a spatially and temporally smooth approximation to its 
position for use in adaptive mesh refinement.  For adaptive mesh refinement
purposes, there may be merit to having a description of the apparent horizon 
in terms of a functional expansion with numerically determined coefficients. 
This functional description immediately emerges from the multipole-moment 
approach.  It could also be numerically determined from the Newton-Raphson
procedure.  With such a development, and with coefficients known as 
smooth functions of time, the horizon can be located on any fine time step
in the vicinity of fine spatial gridding.


Discontinuous Apparent Horizon Movement:

The apparent horizon finder must be employed to search for discontinuous
outward movement of the horizon, as will occur when the two holes 
plunge toward each other.  To handle a discontinuous jump in the apparent
horizon, two strategies are suggested.  One idea is to do a spatial remap 
of the computational domain prior to proceeding with the time evolution.  
The purpose of this remap is to utilize newly available memory which might 
otherwise, at least temporarily, go unused as the new, larger interior 
region is excised.  The disadvantage of this is that it interrupts the 
second-order time evolution with a step whose convergence properties may 
not be well understood.  The alternative and probably preferable idea is 
to continue the time evolution outside the new horizon with the same grid 
structure.  This frees up some memory which can be used later to keep the 
exterior region well-resolved.  From this point on in the evolution, the 
shift may be reformulated to approximate the simpler choices that have
been effective in handling one black hole.<br>


    TESTS AND TIMESCALES:

Cornell/UNC Empire code development and inner boundary test milestones:


Development of the Empire code (parallel, adaptive-mesh implementation of
the new hyperbolic formulation) has revealed some necessary modifications
of DAGH, which are being pursued.  The implementation of the Berger-Oliger 
algorithm with DAGH is still under development.  With these present caveats,
the Cornell/UNC effort will attempt within a few weeks of resolution of 
DAGH problems a convergence test of the Empire code WITHOUT a shift vector.  
We expect to use a linearized wave in the preliminary test and to evolve 
Schwarzschild for a short time with no shift as the second test in order 
to verify the convergence properties of the non-linear terms.

Within the next several months implementation of a non-adaptive version of 
a causally stable shift algorithm and dynamic slicing condition will be 
attempted in 3D, based upon results garnered with a 1D spherical code.  
Choosing a "good" shift is decoupled from the successful implementation of 
a shift algorithm.  Identifying a "good" shift may require longer term
development.

Once a stable, causal shift and dynamic slicing condition has been succesfully
tested, a non-adaptive test of a single hole translating across the grid can 
be attempted.  This test is seen as a significant milestone in the 
development of a code that will simulate the orbital dynamics of two holes.
Estimating when this test might be accomplished is a highly uncertain task
but the goal is make the attempt before the end of the summer.

Potential problems include memory limitations that may prevent high 
accuracy convergence tests for the near future.  At present, 128x128x128 
grids seem to be about the limit for a uniform grid and this may not be 
sufficient resolution.  The timescale for resolution of problems with DAGH 
is also a source of uncertainty.


NCSA 3D code development and inner boundary test milestones:


A summary of comparable efforts at NCSA in testing apparent horizon
inner boundaries with their 3D codes will be summarized in a later 
issue.