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\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ***** macro definitions etc *****
%
\bibliographystyle{alpha}
% misc text stuff
\def\code#1{{\tt #1}} % for formatting code
\def\ie{\hbox{i.e.}}
\def\ahf{\code{AHFinderDirect}} % our own name
% get size/spacing of "++" right, cf online C++ FAQ question 35.1
\def\Cplusplus{\hbox{C\raise.25ex\hbox{\footnotesize ++}}}
% misc math mode stuff
\def\const{{\rm const}}
\def\ij{_{ij}}
\def\upij{^{ij}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ***** title/author/abstract *****
%
\begin{document}
\title{The \ahf{} Thorn}
\author{Jonathan Thornburg\quad{}\code{<jthorn@aei.mpg.de>}}
%
% We want CVS to expand the Id keyword on the next line, but we don't
% want TeX to go into math mode to typeset the expansion (because that
% wouldn't look as nice in the output), so we use the "$ $" construct
% to get TeX out of math mode again when typesetting the expansion.
%
\date{$ $Id$ $}
\maketitle
\abstract{
This document describes the \ahf{} thorn. This thorn locates
apparent horizons in a numerically computed slice using a direct
method, posing the apparent horizon equation as an elliptic PDE
on angular-coordinate space. This is very fast, but requires a
``reasonable'' initial guess.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementation Notes}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Overview}
The \ahf{} thorn is written primarily in \Cplusplus{}, calling
C and Fortran~77 numerical libraries. \Cplusplus{} has proven to be
a powerful and flexible language for this type of programming, and
I think this thorn (particularly the interpatch interpolation) would
have been significantly harder to write in a lower-level language
like C or Fortran~77.
The implementation is only loosely coupled to Cactus -- in general
the code uses its own types and classes, which are implemented on
top of the Cactus ones. Notably, all floating point arithmetic is
done using the type \code{fp}, a typedef for \code{CCTK\_REAL}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Portability}
The code should be fairly portable to modern \Cplusplus{} compilers.
Templates are used, but only template classes templated on floating-point
or integer types, and these templates are always instantiated explicitly.
\code{bool}, \code{mutable}, and \code{typename} are used, as are the
new \code{for}-loop scope rules. The code has been compiled and run
successfully using gcc~2.95.2 on x86 GNU/Linux systems and using
Digital C~V5.6 on Digital Unix~V4.0. The code won't compile using
badly broken compilers like the current (mid-2001) version of Microsoft
Visual \Cplusplus.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\Cplusplus{} Code}
The main high-level \Cplusplus{} classes in the implementation are
as follows:
\begin{description}
\item[\code{patch\_system}]
is the top-level class representing a multipatch system.
\item[\code{x\_patch}, \code{y\_patch}, \code{z\_patch}]
represent individual patches near the $x-$, $y-$, and $z-$axes
respectively.
\item[\code{patch}]
is an virtual base class representing a grid patch.
\item[\code{patch\_edge}]
represents the geometry of a single edge of a patch,
\ie{} it represents the conversions between (perpendicular,parallel)
and (rho,sigma) coordinates.
\item[\code{ghost\_data}]
\item[\code{patch\_interp}]
\item[\code{interpatch\_ghost\_data}]
\item[\code{symmetry\_ghost\_data}]
\end{description}
Class \code{patch} is in term implemented using the following
helper classes:
\begin{description}
\item[\code{fd\_grid}]
\item[\code{grid}]
\item[\code{grid\_arrays}]
\end{description}
The implementation also uses the following low-level generic helper
classes:
\begin{description}
\item[\code{array<fp>}]
is a template class (templated on the integer or floating-point
type of the array elements) providing multidimensional arrays.
At present these are stored directly in C++ \code{new[]}-allocated
storage, but sometime soon I'll add an option to use Cactus
grid functions or local arrays.
\item[\code{linear\_map<fp>}]
is a template class (templated on the floating-point type)
representing a linear map between a contiguous interval of
integers, and floating-point numbers.
\item[\code{cpm\_map}]
represents a mapping from the integers to the integers,
of the form $i \rightarrow \const \pm i$; the name
abbreviates ``\underline{c}onstant \underline{p}lus or
\underline{m}inus'' map.
\item[\code{fuzzy<fp>}]
is a template class (templated on the floating-point type)
providing fuzzy arithmetic, to try to paper over some of the
effects of floating-point rounding errors. For example,
one can write
\begin{verbatim}
for (fp x = 0.0 ; fuzzy<fp>::LE(x,1.0) ; x += 0.1)
{
// ...
}
\end{verbatim}
and have the loop execute as one would naievly expect,
even if rounding errors cause the final value of \code{x}
to be 0.9999999999999999 or 1.000000000000001 or suchlike.
\item[\code{round<fp>}]
\end{description}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maple Code}
The relativity code is all machine-generated from Maple code, which
also uses a Maple preprocessor I wrote in Perl.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{jt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
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