/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.11 by adcroft, Tue Nov 13 19:01:42 2001 UTC revision 1.20 by jmc, Thu Jul 7 21:36:34 2005 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sec:tracer_equations}  \label{sect:tracer_equations}
6    \begin{rawhtml}
7    <!-- CMIREDIR:tracer_equations: -->
8    \end{rawhtml}
9    
10  The basic discretization used for the tracer equations is the second  The basic discretization used for the tracer equations is the second
11  order piece-wise constant finite volume form of the forced  order piece-wise constant finite volume form of the forced
# Line 15  part of the tracer equations and the var Line 18  part of the tracer equations and the var
18  described here.  described here.
19    
20  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
21  \label{sec:tracer_equations_abII}  \label{sect:tracer_equations_abII}
22    \begin{rawhtml}
23    <!-- CMIREDIR:tracer_equations_abII: -->
24    \end{rawhtml}
25    
26  The default advection scheme is the centered second order method which  The default advection scheme is the centered second order method which
27  requires a second order or quasi-second order time-stepping scheme to  requires a second order or quasi-second order time-stepping scheme to
# Line 123  the forward method. Line 129  the forward method.
129    
130    
131  \section{Linear advection schemes}  \section{Linear advection schemes}
132    \label{sect:tracer-advection}
133    \begin{rawhtml}
134    <!-- CMIREDIR:linear_advection_schemes: -->
135    \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
# Line 349  if the limiter is set to zero. Line 359  if the limiter is set to zero.
359    
360    
361  \section{Non-linear advection schemes}  \section{Non-linear advection schemes}
362    \begin{rawhtml}
363    <!-- CMIREDIR:non-linear_advection_schemes: -->
364    \end{rawhtml}
365    
366  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
367  widely used in computational fluid dynamics (non-linear does not refer  widely used in computational fluid dynamics (non-linear does not refer
# Line 449  to centered second order advection in th Line 462  to centered second order advection in th
462    
463  The DST3 method described above must be used in a forward-in-time  The DST3 method described above must be used in a forward-in-time
464  manner and is stable for $0 \le |c| \le 1$. Although the scheme  manner and is stable for $0 \le |c| \le 1$. Although the scheme
465  appears to be forward-in-time, it is in fact second order in time and  appears to be forward-in-time, it is in fact third order in time and
466  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
467  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
468  Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for  Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for
# Line 653  $W$: {\bf rTrans} (local) Line 666  $W$: {\bf rTrans} (local)
666    
667    
668  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
669    \label{sect:tracer_advection_schemes}
670    \begin{rawhtml}
671    <!-- CMIREDIR:comparison_of_advection_schemes: -->
672    \end{rawhtml}
673    
674    \begin{table}[htb]
675    \centering
676     \begin{tabular}[htb]{|l|c|c|c|c|l|}
677       \hline
678       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
679                        &      & A.B. & dimension & (1 dim) & \\
680       \hline \hline
681       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
682       \hline
683       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
684       \hline
685       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
686       \hline \hline
687    %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\
688    %  \hline
689       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
690       \hline \hline
691       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
692       \hline
693       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
694       \hline
695     \end{tabular}
696     \caption{Summary of the different advection schemes available in MITgcm.
697              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
698              The code corresponds to the number used to select the corresponding
699              advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
700              file {\em data} selects the $3^{rd}$ order upwind advection scheme
701              for temperature).
702       }
703     \label{tab:advectionShemes_summary}
704    \end{table}
705    
706    
707  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and
708  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
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