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Diffstat (limited to 'doc')
-rw-r--r-- | doc/documentation.tex | 43 |
1 files changed, 31 insertions, 12 deletions
diff --git a/doc/documentation.tex b/doc/documentation.tex index 69a6593..8e9e4b3 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -35,10 +35,14 @@ keyword parameter {\tt time::timestep\_method}. \begin{itemize} -\item{} {\tt time::timestep\_method = ``given''} The timestep is fixed to the +\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 +\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}. @@ -48,25 +52,37 @@ $$ Note that it is up to the user to custom {\tt dtfac} to take into account the dimension of the space being used, and the wave speed. -\item{} {\tt time::timestep\_method = ``courant\_speed''} This choice implements a +\item{} {\tt time::timestep\_method = "courant\_speed"} + + This choice implements a dynamic courant type condition, the timestep being set before each iteration using the spatial dimension of the grid, the spatial grid sizes, the parameter {\tt courant\_fac} and the grid variable {\tt courant\_wave\_speed}. The algorithm used is $$ -\Delta t = \mbox{\tt courant\_fac} * \mbox{min}(\Delta x^i)/\mbox{courant\_wave\_speed}/\sqrt(\mbox{dim}) +\Delta t = \mbox{\tt courant\_fac} * \mbox{min}(\Delta x^i)/\mbox{\tt courant\_wave\_speed}/\sqrt{\mbox dim} $$ For this algorithm to be successful, the variable {\tt courant\_wave\_speed} - must have been set by a thorn to the maximum wave speed on the grid. + must have been set by some thorn to the maximum propagation speed on the grid {\it before} + this thorn sets the timestep, that is {\tt AT POSTSTEP BEFORE Time\_Courant} (or earlier + in the evolution loop). [Note: The name {\tt courant\_wave\_speed} was poorly + chosen here, the required speed is the maximum propagation speed on + the grid which may be larger than the maximum wave speed (for example + with a shock wave in hydrodynamics, also it is possible to have + propagation without waves as with a pure advection equation). + +\item{} {\tt time::timestep\_method = "courant\_time"} -\item{} {\tt time::timestep\_method = ``courant\_time''} This choice is similar to the + This choice is similar to the method {\tt courant\_speed} above, in implementing a dynamic timestep. However the timestep is chosen using $$ -\Delta t = \mbox{\tt courant\_fac} * \mbox{\tt courant\_min\_time}/\sqrt(\mbox{dim}) +\Delta t = \mbox{\tt courant\_fac} * \mbox{\tt courant\_min\_time}/\sqrt{\mbox dim} $$ - where the grid variable {\tt courant\_min\_time} must be set by a thorn to - the minimum time for a wave to cross a gridzone. + where the grid variable {\tt courant\_min\_time} must be set by some thorn to + the minimum time for a wave to cross a gridzone {\it before} + this thorn sets the timestep, that is {\tt AT POSTSTEP BEFORE Time\_Courant} (or earlier + in the evolution loop). \end{itemize} @@ -78,23 +94,26 @@ Note that for hyperbolic problems, the Courant condition gives a minimum requirement for stability, namely that the numerical domain of dependency must encompass the physical domain of dependency, or $$ -\Delta t \le \mbox{min}(\Delta x^i)/\mbox{wave speed}/\sqrt(\mbox{dim}) +\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_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 @@ -102,7 +121,7 @@ The following parameters set the timestep to be 0.25 grid::dx = 0.5 grid::dy = 1.0 grid::dz = 1.0 -time::timestep_method = ``courant_static'' +time::timestep_method = "courant_static" time::dtfac = 0.5 \end{verbatim} } |