/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.14 by adcroft, Wed Nov 14 21:07:13 2001 UTC revision 1.16 by edhill, Thu Aug 7 18:27:52 2003 UTC
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1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4    This chapter lays out the numerical schemes that are
5    employed in the core MITgcm algorithm. Whenever possible
6    links are made to actual program code in the MITgcm implementation.
7    The chapter begins with a discussion of the temporal discretization
8    used in MITgcm. This discussion is followed by sections that
9    describe the spatial discretization. The schemes employed for momentum
10    terms are described first, afterwards the schemes that apply to
11    passive and dynamically active tracers are described.
12    
13    
14    \section{Time-stepping}
15  The equations of motion integrated by the model involve four  The equations of motion integrated by the model involve four
16  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and
17  salt/moisture, $S$, and three diagnostic equations for vertical flow,  salt/moisture, $S$, and three diagnostic equations for vertical flow,
# Line 70  temporary.} Line 81  temporary.}
81  \begin{figure}  \begin{figure}
82  \begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing}  \begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing}
83  aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill  aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill
84  FORWARD\_STEP \\  \filelink{FORWARD\_STEP}{model-src-forward_step.F} \\
85  \> DYNAMICS \\  \> DYNAMICS \\
86  \>\> TIMESTEP \` $u^*$,$v^*$ (\ref{eq:ustar-rigid-lid},\ref{eq:vstar-rigid-lid}) \\  \>\> TIMESTEP \` $u^*$,$v^*$ (\ref{eq:ustar-rigid-lid},\ref{eq:vstar-rigid-lid}) \\
87  \> SOLVE\_FOR\_PRESSURE \\  \> SOLVE\_FOR\_PRESSURE \\
# Line 80  FORWARD\_STEP \\ Line 91  FORWARD\_STEP \\
91  \>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\  \>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\
92  \>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid})  \>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid})
93  \end{tabbing} \end{minipage} } \end{center}  \end{tabbing} \end{minipage} } \end{center}
94  \caption{Calling tree for the pressure method algorihtm}  \caption{Calling tree for the pressure method algorithm
95      (\filelink{FORWARD\_STEP}{model-src-forward_step.F})}
96  \label{fig:call-tree-pressure-method}  \label{fig:call-tree-pressure-method}
97  \end{figure}  \end{figure}
98    
# Line 162  The correspondence to the code is as fol Line 174  The correspondence to the code is as fol
174  \item  \item
175  the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid},  the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid},
176  stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in  stepping forward $u^n$ and $v^n$ to $u^{*}$ and $v^{*}$ is coded in
177  {\em TIMESTEP.F}  \filelink{TIMESTEP()}{model-src-timestep.F}
178  \item  \item
179  the vertical integration, $H \widehat{u^*}$ and $H  the vertical integration, $H \widehat{u^*}$ and $H
180  \widehat{v^*}$, divergence and inversion of the elliptic operator in  \widehat{v^*}$, divergence and inversion of the elliptic operator in
181  equation \ref{eq:elliptic} is coded in {\em  equation \ref{eq:elliptic} is coded in
182  SOLVE\_FOR\_PRESSURE.F}  \filelink{SOLVE\_FOR\_PRESSURE()}{model-src-solve_for_pressure.F}
183  \item  \item
184  finally, the new flow field at time level $n+1$ given by equations  finally, the new flow field at time level $n+1$ given by equations
185  \ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in {\em CORRECTION\_STEP.F}.  \ref{eq:un+1-rigid-lid} and \ref{eq:vn+1-rigid-lid} is calculated in
186    \filelink{CORRECTION\_STEP()}{model-src-correction_step.F}.
187  \end{itemize}  \end{itemize}
188  The calling tree for these routines is given in  The calling tree for these routines is given in
189  Fig.~\ref{fig:call-tree-pressure-method}.  Fig.~\ref{fig:call-tree-pressure-method}.
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