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% *======================================================================*
% Cactus Thorn template for ThornGuide documentation
% Author: Ian Kelley
% Date: Sun Jun 02, 2002
% $Header$
%
% Thorn documentation in the latex file doc/documentation.tex
% will be included in ThornGuides built with the Cactus make system.
% The scripts employed by the make system automatically include
% pages about variables, parameters and scheduling parsed from the
% relevant thorn CCL files.
%
% This template contains guidelines which help to assure that your
% documentation will be correctly added to ThornGuides. More
% information is available in the Cactus UsersGuide.
%
% Guidelines:
% - Do not change anything before the line
% % START CACTUS THORNGUIDE",
% except for filling in the title, author, date, etc. fields.
% - Each of these fields should only be on ONE line.
% - Author names should be separated with a \\ or a comma.
% - You can define your own macros, but they must appear after
% the START CACTUS THORNGUIDE line, and must not redefine standard
% latex commands.
% - To avoid name clashes with other thorns, 'labels', 'citations',
% 'references', and 'image' names should conform to the following
% convention:
% ARRANGEMENT_THORN_LABEL
% For example, an image wave.eps in the arrangement CactusWave and
% thorn WaveToyC should be renamed to CactusWave_WaveToyC_wave.eps
% - Graphics should only be included using the graphicx package.
% More specifically, with the "\includegraphics" command. Do
% not specify any graphic file extensions in your .tex file. This
% will allow us to create a PDF version of the ThornGuide
% via pdflatex.
% - References should be included with the latex "\bibitem" command.
% - Use \begin{abstract}...\end{abstract} instead of \abstract{...}
% - Do not use \appendix, instead include any appendices you need as
% standard sections.
% - For the benefit of our Perl scripts, and for future extensions,
% please use simple latex.
%
% *======================================================================*
%
% Example of including a graphic image:
% \begin{figure}[ht]
% \begin{center}
% \includegraphics[width=6cm]{MyArrangement_MyThorn_MyFigure}
% \end{center}
% \caption{Illustration of this and that}
% \label{MyArrangement_MyThorn_MyLabel}
% \end{figure}
%
% Example of using a label:
% \label{MyArrangement_MyThorn_MyLabel}
%
% Example of a citation:
% \cite{MyArrangement_MyThorn_Author99}
%
% Example of including a reference
% \bibitem{MyArrangement_MyThorn_Author99}
% {J. Author, {\em The Title of the Book, Journal, or periodical}, 1 (1999),
% 1--16. {\tt http://www.nowhere.com/}}
%
% *======================================================================*
% 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
% Do not delete next line
% START CACTUS THORNGUIDE
% 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}
% Do not delete next line
% END CACTUS THORNGUIDE
\end{document}
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