/[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 1.23 by jmc, Tue Jan 15 23:52:12 2008 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 614  as if in one dimension: Line 627  as if in one dimension:
627  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
628  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
629             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
630  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
631  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
632             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
633  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
634  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
635             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
636  \end{eqnarray}  \end{eqnarray}
# Line 651  $W$: {\bf rTrans} (local) Line 664  $W$: {\bf rTrans} (local)
664    
665  \end{minipage} }  \end{minipage} }
666    
667    \begin{figure}
668    \resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}}
669    \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
670    \label{fig:advect-multidim_cs}
671    }
672    \end{figure}
673    
674  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
675    \label{sect:tracer_advection_schemes}
676    \begin{rawhtml}
677    <!-- CMIREDIR:comparison_of_advection_schemes: -->
678    \end{rawhtml}
679    
680    \begin{table}[htb]
681    \centering
682     \begin{tabular}[htb]{|l|c|c|c|c|l|}
683       \hline
684       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
685                        &      & A.B. & dimension & (1 dim) & \\
686       \hline \hline
687       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
688       \hline
689       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
690       \hline
691       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
692       \hline
693       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
694       \hline \hline
695       $2^{nd}$order DST (Lax-Wendroff)  & 20 &
696                             No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
697       \hline
698       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
699       \hline \hline
700       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
701       \hline
702       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
703       \hline
704     \end{tabular}
705     \caption{Summary of the different advection schemes available in MITgcm.
706              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
707              The code corresponds to the number used to select the corresponding
708              advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
709              file {\em data} selects the $3^{rd}$ order upwind advection scheme
710              for temperature).
711       }
712     \label{tab:advectionShemes_summary}
713    \end{table}
714    
715    
716  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
717  \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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