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\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}
\title{Time}
\author{Gabrielle Allen}
\date{$ $Date$ $}
\maketitle
% Do not delete next line
% START CACTUS THORNGUIDE
\begin{abstract}
Calculates the timestep used for an evolution
\end{abstract}
\section{Purpose}
This thorn provides 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}.
\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
iteration 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
$$
\Delta t = \mbox{\tt courant\_fac} * \mbox{min}(\Delta x^i)/\mbox{\tt courant\_wave\_speed}/\sqrt{\mbox dim}
$$
For this algorithm to be successful, the variable {\tt courant\_wave\_speed}
must have been set by some thorn to the maximum propagation speed on the grid {\it before}
this thorn sets the timestep, that is {\tt AT POSTSTEP BEFORE Time\_Courant} (or earlier
in the evolution loop). [Note: The name {\tt courant\_wave\_speed} was poorly
chosen here, the required speed is the maximum propagation speed on
the grid which may be larger than the maximum wave speed (for example
with a shock wave in hydrodynamics, also it is possible to have
propagation without waves as with a pure advection equation).
\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 some thorn to
the minimum time for a wave to cross a gridzone {\it before}
this thorn sets the timestep, that is {\tt AT POSTSTEP BEFORE Time\_Courant} (or earlier
in the evolution loop).
\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
$$
\Delta t \le \mbox{min}(\Delta x^i)/\mbox{wave speed}/\sqrt{\mbox dim}
$$
\section{Examples}
\noindent
{\bf Fixed Value Timestep}
{\tt
\begin{verbatim}
time::timestep_method = "given"
time::timestep = 0.1
\end{verbatim}
}
\noindent
{\bf Calculate Static Timestep Based on Grid Spacings}
\noindent
The following parameters set the timestep to be 0.25
{\tt
\begin{verbatim}
grid::dx = 0.5
grid::dy = 1.0
grid::dz = 1.0
time::timestep_method = "courant_static"
time::dtfac = 0.5
\end{verbatim}
}
% Do not delete next line
% END CACTUS THORNGUIDE
\end{document}
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