/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.18 by jmc, Thu Oct 14 22:22:30 2004 UTC revision 1.19 by edhill, Sat Oct 16 03:40:12 2004 UTC
# Line 12  passive and dynamically active tracers a Line 12  passive and dynamically active tracers a
12    
13    
14  \section{Time-stepping}  \section{Time-stepping}
15    \begin{rawhtml}
16    <!-- CMIREDIR:time-stepping: -->
17    \end{rawhtml}
18    
19  The equations of motion integrated by the model involve four  The equations of motion integrated by the model involve four
20  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and  prognostic equations for flow, $u$ and $v$, temperature, $\theta$, and
21  salt/moisture, $S$, and three diagnostic equations for vertical flow,  salt/moisture, $S$, and three diagnostic equations for vertical flow,
# Line 61  treated more exactly, including non-line Line 65  treated more exactly, including non-line
65  described in section \ref{sect:nonlinear-freesurface}.  described in section \ref{sect:nonlinear-freesurface}.
66    
67    
68  \section{Pressure method with rigid-lid} \label{sect:pressure-method-rigid-lid}  \section{Pressure method with rigid-lid}
69    \label{sect:pressure-method-rigid-lid}
70    \begin{rawhtml}
71    <!-- CMIREDIR:pressure_method_rigid_lid: -->
72    \end{rawhtml}
73    
74  \begin{figure}  \begin{figure}
75  \begin{center}  \begin{center}
# Line 203  G_u^{(n+1/2)} Line 211  G_u^{(n+1/2)}
211    
212  \section{Pressure method with implicit linear free-surface}  \section{Pressure method with implicit linear free-surface}
213  \label{sect:pressure-method-linear-backward}  \label{sect:pressure-method-linear-backward}
214    \begin{rawhtml}
215    <!-- CMIREDIR:pressure_method_linear_backward: -->
216    \end{rawhtml}
217    
218  The rigid-lid approximation filters out external gravity waves  The rigid-lid approximation filters out external gravity waves
219  subsequently modifying the dispersion relation of barotropic Rossby  subsequently modifying the dispersion relation of barotropic Rossby
# Line 279  pressure-method. Line 290  pressure-method.
290    
291  \section{Explicit time-stepping: Adams-Bashforth}  \section{Explicit time-stepping: Adams-Bashforth}
292  \label{sect:adams-bashforth}  \label{sect:adams-bashforth}
293    \begin{rawhtml}
294    <!-- CMIREDIR:adams_bashforth: -->
295    \end{rawhtml}
296    
297  In describing the the pressure method above we deferred describing the  In describing the the pressure method above we deferred describing the
298  time discretization of the explicit terms. We have historically used  time discretization of the explicit terms. We have historically used
# Line 346  A stability analysis for a relaxation eq Line 360  A stability analysis for a relaxation eq
360    
361    
362  \section{Implicit time-stepping: backward method}  \section{Implicit time-stepping: backward method}
363    \begin{rawhtml}
364    <!-- CMIREDIR:implicit_time-stepping_backward: -->
365    \end{rawhtml}
366    
367  Vertical diffusion and viscosity can be treated implicitly in time  Vertical diffusion and viscosity can be treated implicitly in time
368  using the backward method which is an intrinsic scheme.  using the backward method which is an intrinsic scheme.
# Line 391  implicit and are thus cast as a an expli Line 408  implicit and are thus cast as a an expli
408    
409  \section{Synchronous time-stepping: variables co-located in time}  \section{Synchronous time-stepping: variables co-located in time}
410  \label{sect:adams-bashforth-sync}  \label{sect:adams-bashforth-sync}
411    \begin{rawhtml}
412    <!-- CMIREDIR:adams_bashforth_sync: -->
413    \end{rawhtml}
414    
415  \begin{figure}  \begin{figure}
416  \begin{center}  \begin{center}
# Line 508  time-step. The corresponding calling tre Line 528  time-step. The corresponding calling tre
528    
529  \section{Staggered baroclinic time-stepping}  \section{Staggered baroclinic time-stepping}
530  \label{sect:adams-bashforth-staggered}  \label{sect:adams-bashforth-staggered}
531    \begin{rawhtml}
532    <!-- CMIREDIR:adams_bashforth_staggered: -->
533    \end{rawhtml}
534    
535  \begin{figure}  \begin{figure}
536  \begin{center}  \begin{center}
# Line 650  time-level variables and terms correspon Line 673  time-level variables and terms correspon
673    
674  \section{Non-hydrostatic formulation}  \section{Non-hydrostatic formulation}
675  \label{sect:non-hydrostatic}  \label{sect:non-hydrostatic}
676    \begin{rawhtml}
677    <!-- CMIREDIR:non-hydrostatic_formulation: -->
678    \end{rawhtml}
679    
680  The non-hydrostatic formulation re-introduces the full vertical  The non-hydrostatic formulation re-introduces the full vertical
681  momentum equation and requires the solution of a 3-D elliptic  momentum equation and requires the solution of a 3-D elliptic
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