From 9075623aba80099abe9e32f220586829bd6b4986 Mon Sep 17 00:00:00 2001 From: rhaas Date: Fri, 22 Mar 2013 16:50:09 +0000 Subject: expand documentation this was kindly provided by Bernard Kelly, who writes: I've added to the documentation for the Dissipation thorn, since what was there was just a pointer to the Kreiss & Oliger paper. Now there's an expression for what actually happens in the thorn, and some discussion of the main parameters. git-svn-id: http://svn.cactuscode.org/arrangements/CactusNumerical/Dissipation/trunk@57 850bcc8b-0e4f-0410-8c26-8d28fbf1eda9 --- doc/documentation.tex | 43 +++++++++++++++++++++++++++++++++++++++---- 1 file changed, 39 insertions(+), 4 deletions(-) diff --git a/doc/documentation.tex b/doc/documentation.tex index ecc3634..e1b5354 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -79,7 +79,7 @@ \begin{document} % The author of the documentation -\author{Erik Schnetter \textless schnetter@aei.mpg.de\textgreater} +\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} @@ -98,7 +98,7 @@ % Add an abstract for this thorn's documentation \begin{abstract} -Add fourth order Kreiss-Oliger dissipation to the right hand side of +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} @@ -107,8 +107,43 @@ use MoL. % Remove them or add your own. \section{Physical System} -For a description of the artificial dissipation, see -\cite{kreiss-oliger}. +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 -- cgit v1.2.3