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%  Date: Sun Jun 02, 2002
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% If you are using CVS use this line to give version information
% $Header$

\documentclass{article}

% Use the Cactus ThornGuide style file
% (Automatically used from Cactus distribution, if you have a 
%  thorn without the Cactus Flesh download this from the Cactus
%  homepage at www.cactuscode.org)
\usepackage{../../../../doc/latex/cactus}

\begin{document}

% The author of the documentation
\author{Erik Schnetter \textless schnetter@aei.mpg.de\textgreater, Bernard Kelly \textless bernard.j.kelly@nasa.gov\textgreater}

% The title of the document (not necessarily the name of the Thorn)
\title{Dissipation}

% the date your document was last changed, if your document is in CVS, 
% please use:
\date{$ $Date$ $}

\maketitle

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% Add all definitions used in this documentation here 
%   \def\mydef etc

% Add an abstract for this thorn's documentation
\begin{abstract}
Add $n$th-order Kreiss-Oliger dissipation to the right hand side of
evolution equations.  This thorn is intended for time evolutions that
use MoL.
\end{abstract}

% The following sections are suggestive only.
% Remove them or add your own.

\section{Physical System}
For a description of Kreiss-Oliger artificial dissipation, see \cite{kreiss-oliger}.

The additional dissipation terms appear as follows, for a general grid function $U$. Here, the
tensor character of the field is irrelevant: each component of, say, $\tilde{\gamma}_{ij}$ is
treated as an independent field for dissipation purposes.
%
\begin{eqnarray*}
\partial_t U &=& \partial_t U + (-1)^{(p+3)/2} \epsilon \frac{1}{2^{p+1}} \left( h_x^{p} \frac{\partial^{(p+1)}}{\partial x^{(p+1)}} + h_y^{p} \frac{\partial^{(p+1)}}{\partial y^{(p+1)}} +
h_z^{p} \frac{\partial^{(p+1)}}{\partial z^{(p+1)}}\right) U, \\
             &=& \partial_t U + (-1)^{(p+3)/2} \epsilon \frac{h^{p}}{2^{p+1}} \left( \frac{\partial^{(p+1)}}{\partial x^{(p+1)}} + \frac{\partial^{(p+1)}}{\partial y^{(p+1)}} +
\frac{\partial^{(p+1)}}{\partial z^{(p+1)}}\right) U,
\end{eqnarray*}
%
where $h_x$, $h_y$, and $h_z$ are the local grid spacings in each Cartesian direction, and the
second equality holds in the usual situation where the three are equal: $h_x = h_y = h_z = h$.

\section{Implementation in Cactus}

The \texttt{Dissipation} thorn's dissipation rate is controlled by a small number of parameters:
%
\begin{itemize}
  \item \texttt{order} is the order $p$ of the dissipation, implying the use of the $(p+1)$-st spatial derivatives;
  \item \texttt{epsdiss} is the overall dissipation strength $\epsilon$.
\end{itemize}

Currently available values of \texttt{order} are $p \in \{1, 3, 5, 7, 9\}$. To apply dissipation at
order $p$ requires that we have at least $(p+1)/2$ ghostzones --- $\{1, 2, 3, 4, 5\}$, respectively.

The list of fields to be dissipated is specified in the parameter \texttt{vars}. The thorn does
not allow for individually tuned dissipation strengths for different fields. However, the
dissipation strength $\epsilon$ can be varied according to refinement level, using the parameter
array \texttt{epsdis\_for\_level}, which overrides \texttt{epsdiss} if set.

The thorn also allows for enhanced dissipation within the apparent horizons, triggered by the
boolean parameter \texttt{extra\_dissipation\_in\_horizons}, and near the outer boundary,
triggered by the boolean parameter \texttt{extra\_dissipation\_at\_outerbound}. Both of these
default to ``no''.

\subsection{Acknowledgements}
I thank Scott Hawley who wrote a very similar thorn
\texttt{HawleyThorns/Dissipation} for evolutions that do not use MoL;
this thorn here is modelled after his.

\begin{thebibliography}{9}
\bibitem{kreiss-oliger}
H. Kreiss and J. Oliger, \emph{Methods for the Approximate Solution of
Time Dependent Problems}, vol.\ 10 of Global Atmospheric Research
Programme (GARP): GARP Publication Series (GARP Publication, 1973)
\end{thebibliography}

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\end{document}