/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.20 by jmc, Thu Jul 7 21:36:34 2005 UTC revision 1.26 by jmc, Mon Aug 30 23:09:19 2010 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sect:tracer_equations}  \label{sec:tracer_equations}
6  \begin{rawhtml}  \begin{rawhtml}
7  <!-- CMIREDIR:tracer_equations: -->  <!-- CMIREDIR:tracer_equations: -->
8  \end{rawhtml}  \end{rawhtml}
# Line 18  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{sect:tracer_equations_abII}  \label{sec:tracer_equations_abII}
22  \begin{rawhtml}  \begin{rawhtml}
23  <!-- CMIREDIR:tracer_equations_abII: -->  <!-- CMIREDIR:tracer_equations_abII: -->
24  \end{rawhtml}  \end{rawhtml}
# Line 129  the forward method. Line 129  the forward method.
129    
130    
131  \section{Linear advection schemes}  \section{Linear advection schemes}
132  \label{sect:tracer-advection}  \label{sec:tracer-advection}
133  \begin{rawhtml}  \begin{rawhtml}
134  <!-- CMIREDIR:linear_advection_schemes: -->  <!-- CMIREDIR:linear_advection_schemes: -->
135  \end{rawhtml}  \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}}
139  \caption{  \caption{
140  Comparison of 1-D advection schemes. Courant number is 0.05 with 60  Comparison of 1-D advection schemes. Courant number is 0.05 with 60
141  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 153  $\mu=c/(1-c)$. Line 153  $\mu=c/(1-c)$.
153  \end{figure}  \end{figure}
154    
155  \begin{figure}  \begin{figure}
156  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}}
157  \caption{  \caption{
158  Comparison of 1-D advection schemes. Courant number is 0.89 with 60  Comparison of 1-D advection schemes. Courant number is 0.89 with 60
159  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 359  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{sec:non-linear_advection_schemes}
363  \begin{rawhtml}  \begin{rawhtml}
364  <!-- CMIREDIR:non-linear_advection_schemes: -->  <!-- CMIREDIR:non-linear_advection_schemes: -->
365  \end{rawhtml}  \end{rawhtml}
# Line 549  $\tau$: {\bf tracer} (argument) Line 550  $\tau$: {\bf tracer} (argument)
550  \subsection{Multi-dimensional advection}  \subsection{Multi-dimensional advection}
551    
552  \begin{figure}  \begin{figure}
553  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}
554  \caption{  \caption{
555  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
556  of a resolved Gaussian feature. Courant number is 0.01 with  of a resolved Gaussian feature. Courant number is 0.01 with
# Line 570  lines). Line 571  lines).
571  \end{figure}  \end{figure}
572    
573  \begin{figure}  \begin{figure}
574  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}
575  \caption{  \caption{
576  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
577  of a resolved Gaussian feature. Courant number is 0.27 with  of a resolved Gaussian feature. Courant number is 0.27 with
# Line 591  lines). Line 592  lines).
592  \end{figure}  \end{figure}
593    
594  \begin{figure}  \begin{figure}
595  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}
596  \caption{  \caption{
597  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
598  of a resolved Gaussian feature. Courant number is 0.47 with  of a resolved Gaussian feature. Courant number is 0.47 with
# Line 627  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 664  $W$: {\bf rTrans} (local) Line 665  $W$: {\bf rTrans} (local)
665    
666  \end{minipage} }  \end{minipage} }
667    
668    \begin{figure}
669    \resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/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}  \label{sec:tracer_advection_schemes}
677  \begin{rawhtml}  \begin{rawhtml}
678  <!-- CMIREDIR:comparison_of_advection_schemes: -->  <!-- CMIREDIR:comparison_of_advection_schemes: -->
679  \end{rawhtml}  \end{rawhtml}
# Line 678  $W$: {\bf rTrans} (local) Line 685  $W$: {\bf rTrans} (local)
685     Advection Scheme & code & use  & use Multi- & Stencil & comments \\     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
686                      &      & A.B. & dimension & (1 dim) & \\                      &      & A.B. & dimension & (1 dim) & \\
687     \hline \hline     \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 \\     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
691     \hline     \hline
692     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
693     \hline     \hline
694     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
695     \hline \hline     \hline \hline
696  %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\     $2^{nd}$order DST (Lax-Wendroff)  & 20 &
697  %  \hline                           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\\     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
700     \hline \hline     \hline \hline
701     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
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