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\documentclass{article}
\begin{document}
\title{Time}
\author{Gabrielle Allen}
\date{1999}
\maketitle
\abstract{Calculates the timestep used for an evolution}
\section{Purpose}
This thorn provides a routines for calculating
the timestep for an evolution based on the spatial Cartesian grid spacing and
a wave speed.
\section{Description}
Thorn {\tt Time} uses one of four methods to decide on the timestep
to be used for the simulation. The method is chosen using the
keyword parameter {\tt time::timestep\_method}. (Note: In releases Beta 8 and
earlier the parameter used was {\tt time::courant\_method}
\begin{itemize}
\item{} {\tt time::timestep\_method = ``given''} The timestep is fixed to the
value of the parameter {\tt time::timestep}.
\item{} {\tt time::timestep\_method = ``courant\_static''} This is the default
method, which calculates the timestep once at the start of the
simulation, based on a simple courant type condition using
the spatial gridsizes and the parameter {\tt time::dtfac}.
$$
\Delta t = \mbox{\tt dtfac} * \mbox{min}(\Delta x^i)
$$
Note that it is up to the user to custom {\tt dtfac} to take
into account the dimension of the space being used, and the wave speed.
\item{} {\tt time::timestep\_method = ``courant\_speed''} This choice implements a
dynamic courant type condition, the timestep being set before each
timestep using the spatial dimension of the grid, the spatial grid sizes, the
parameter {\tt courant\_fac} and the grid variable {\tt courant\_wave\_speed}.
The algorithm used is
$$
\Delta t = \mbox{\tt courant\_fac} * \mbox{min}(\Delta x^i)/\mbox{courant\_wave\_speed}/\sqrt(\mbox{dim})
$$
For this algorithm to be successful, the variable {\tt courant\_wave\_speed}
must have been set by a thorn to the maximum wave speed on the grid.
\item{} {\tt time::timestep\_method = ``courant\_time''} This choice is similar to the
method {\tt courant\_speed} above, in implementing a dynamic timestep.
However the timestep is chosen using
$$
\Delta t = \mbox{\tt courant\_fac} * \mbox{\tt courant\_min\_time}/\sqrt(\mbox{dim})
$$
where the grid variable {\tt courant\_min\_time} must be set by a thorn to
the minimum time for a wave to cross a gridzone.
\end{itemize}
In all cases, Thorn {\tt Time} sets the Cactus variable {\tt cctk\_delta\_time}
which is passed as part of the macro {\tt CCTK\_ARGUMENTS} to thorns called
by the scheduler.
Note that for hyperbolic problems, the Courant condition gives a minimum
requirement for stability, namely that the numerical domain of dependency
must encompass the physical domain of dependency, or
$$
\Delta t \le \mbox{min}(\Delta x^i)/\mbox{wave speed}/\sqrt(\mbox{dim})
$$
\end{itemize}
% Automatically created from the ccl files by using gmake thorndoc
\include{interface}
\include{param}
\include{schedule}
\end{document}
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