/[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 1.24 by cnh, Thu Jan 17 17:42:11 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    \label{sect:non-linear_advection_schemes}
363    \begin{rawhtml}
364    <!-- CMIREDIR:non-linear_advection_schemes: -->
365    \end{rawhtml}
366    
367  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
368  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 463  to centered second order advection in th
463    
464  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
465  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
466  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
467  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
468  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
469  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 628  as if in one dimension:
628  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
629  - \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})
630             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
631  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
632  - \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})
633             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
634  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
635  - \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})
636             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
637  \end{eqnarray}  \end{eqnarray}
# Line 651  $W$: {\bf rTrans} (local) Line 665  $W$: {\bf rTrans} (local)
665    
666  \end{minipage} }  \end{minipage} }
667    
668    \begin{figure}
669    \resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}}
670    \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
671    \label{fig:advect-multidim_cs}
672    }
673    \end{figure}
674    
675  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
676    \label{sect:tracer_advection_schemes}
677    \begin{rawhtml}
678    <!-- CMIREDIR:comparison_of_advection_schemes: -->
679    \end{rawhtml}
680    
681    \begin{table}[htb]
682    \centering
683     \begin{tabular}[htb]{|l|c|c|c|c|l|}
684       \hline
685       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
686                        &      & A.B. & dimension & (1 dim) & \\
687       \hline \hline
688       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
689       \hline
690       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
691       \hline
692       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
693       \hline
694       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
695       \hline \hline
696       $2^{nd}$order DST (Lax-Wendroff)  & 20 &
697                             No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
698       \hline
699       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
700       \hline \hline
701       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
702       \hline
703       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
704       \hline
705     \end{tabular}
706     \caption{Summary of the different advection schemes available in MITgcm.
707              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
708              The code corresponds to the number used to select the corresponding
709              advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
710              file {\em data} selects the $3^{rd}$ order upwind advection scheme
711              for temperature).
712       }
713     \label{tab:advectionShemes_summary}
714    \end{table}
715    
716    
717  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
718  \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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