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{\Large Modules in Numerical Relativity } \\
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{\footnotesize Last revision: Oct. 6, 1994}
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\section{Choice of Physical Domain}

\begin{itemize}
\item spherical symmetry (1 spatial dimension); spherical coordinate $(r)$
\item axisymmetry (2 spatial dimensions); spherical polar coordinates
   $(r,\theta)$
\item no symmetry (3 spactial dimensions)
\begin{itemize}
\item cartesian coordinates $(x,y,z)$
\item spherical coordinates $(r,\theta,\phi)$
\item others (i.e. Cadez coordinates)
\end{itemize}
\end{itemize}

\section{Initial Value Problem}

\begin{itemize}
\item specify initial data that satisfies 4 elliptic ``constraint equations''
\begin{itemize}
\item ``copy'' from known analytic solution
\item use elliptic solver to solve equations
\begin{itemize}
\item identify parameters for 2 BH solutions to constraint equations
\end{itemize}
\end{itemize}
\end{itemize}

\section{Cauchy Development (evolution of spatial surfaces)}

\begin{itemize}
\item hyperbolic equations
\item choice of methodology for approximating PDE solutions
\begin{itemize}
\item finite differencing with structured mesh (current ``method of
choice'')
\item finite elements (Regge Calculus)
\end{itemize}
\end{itemize}

\subsection{Gauge Choice}

\begin{itemize}
\item lapse and shift functions must be chosen in order to specify Cauchy
   development
\begin{itemize}
\item $q^{ij}K_{ij}$ = const (Trace(K) = const) gauge
\begin{itemize}
\item this choice of gauge has ``singularity avoidance'' properties,
but the payment is solving an elliptic equation on each spatial surface (that
is, for each time step)
\end{itemize}
\item Harmonic gauge (shift = 0)
\begin{itemize}
\item simplification of evolution equations, but at the cost of possible 
numerical instabilities, grid stretching, etc. 
\end{itemize}
\item Algebraic gauge
\begin{itemize}
\item easy to impliment (no elliptic equation to solve, as in Tr(K) = const),
but is physically hard to justify
\end{itemize}
\end{itemize}
\end{itemize}

\subsection{Constraint Maintenance}
\begin{itemize}
\item although analytic evolution maintains the constraint equations
(they stay solved throughout the evolution), errors introduced by finite
differencing make the constraints ``stray'' during evolution.
\begin{itemize}
\item method for constraint enforcement (if at all) 
\end{itemize}
\end{itemize}

\subsection{Tracking Apparent Horizons (Inner Boundary)}
\begin{itemize} 
\item for some (but not all!!) coordinate choices, there are
inner boundaries on the computational grid 
\begin{itemize}
\item put the inner boundary ``a little beyond'' apparent horizons; this means
you have to track apparent horizons during evolution -- this can be
tricky.
\end{itemize}
\end{itemize}

\subsection{Addaptive Meshes, Multigrid}
\begin{itemize}
\item criterion for adaptation
\end{itemize}

\subsection{Spacelike waveform extraction}
\begin{itemize}
\item it is possible to extract wave information inside the Cauchy
devolopment (in spatial slices)
\begin{itemize}
\item requires large(r) computational grids
\item still have to worry about outer boundary conditions (outgoing waves)
\end{itemize}
\end{itemize}

\section{Interface between Cauchy Development and Characteristic Surface
Evolution}
\begin{itemize}
\item {\Large IMPORTANT: } must suppress numerically induced reflections
while allowing physical backscattering (may be difficult)
\end{itemize}

\section{Characteristic Surface Evolution}
\begin{itemize}
\item extract waves relevant to LIGO measurements (certain frequency band)
\end{itemize}


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