/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.17 by jmc, Wed Oct 13 18:50:54 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 516  time-step. The corresponding calling tre Line 539  time-step. The corresponding calling tre
539  \caption{  \caption{
540  A schematic of the explicit Adams-Bashforth and implicit time-stepping  A schematic of the explicit Adams-Bashforth and implicit time-stepping
541  phases of the algorithm but with staggering in time of thermodynamic  phases of the algorithm but with staggering in time of thermodynamic
542  variables with the flow. Explicit thermodynamics tendencies are  variables with the flow.
543  evaluated at time level $n$ as a function of the thermodynamics  Explicit momentum tendencies are evaluated at time level $n-1/2$ as a
544  state at that time level $n$ and flow at time $n+1/2$ (dotted arrow). The  function of the flow field at that time level $n-1/2$.
545  explicit tendency from the previous time level, $n-1$, is used to  The explicit tendency from the previous time level, $n-3/2$, is used to
546  extrapolate tendencies to $n+1/2$ (dashed arrow). This extrapolated  extrapolate tendencies to $n$ (dashed arrow).
547  tendency allows thermo-dynamics variables to be stably integrated  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly
548  forward-in-time to render an estimate ($*$-variables) at the $n+1$  at time level $n$ (vertical arrows) and used with the extrapolated tendencies
549  time level (solid arc-arrow). The implicit-in-time operator ${\cal  to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow).
550  L_{\theta,S}}$ is solved to yield the thermodynamic variables at time  The implicit-in-time operator ${\cal L_{u,v}}$ (vertical arrows) is
551  level $n+1$. These are then used to calculate the hydrostatic  then applied to the previous estimation of the the flow field ($*$-variables)
552  pressure/geo-potential, $\phi_{hyd}$ (vertical arrows). The  and yields to the two velocity components $u,v$ at time level $n+1/2$.
553  hydrostatic pressure gradient is evaluated directly at time level  These are then used to calculate the advection term (dashed arc-arrow)
554  $n+1$ in stepping forward the flow variables from $n+1/2$ to $n+3/2$  of the thermo-dynamics tendencies at time step $n$.
555  (solid arc-arrow). }  The extrapolated thermodynamics tendency, from time level $n-1$ and $n$
556    to $n+1/2$, allows thermodynamic variables to be stably integrated
557    forward-in-time (solid arc-arrow) up to time level $n+1$.
558    }
559  \label{fig:adams-bashforth-staggered}  \label{fig:adams-bashforth-staggered}
560  \end{figure}  \end{figure}
561    
# Line 636  for the non-linear free-surface algorith Line 662  for the non-linear free-surface algorith
662    
663  The only difficulty with this approach is apparent in equation  The only difficulty with this approach is apparent in equation
664  \ref{eq:Gt-n-staggered} and illustrated by the dotted arrow  \ref{eq:Gt-n-staggered} and illustrated by the dotted arrow
665  connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect  connecting $u,v^{n+1/2}$ with $G_\theta^{n}$. The flow used to advect
666  tracers around is not naturally located in time. This could be avoided  tracers around is not naturally located in time. This could be avoided
667  by applying the Adams-Bashforth extrapolation to the tracer field  by applying the Adams-Bashforth extrapolation to the tracer field
668  itself and advecting that around but this approach is not yet  itself and advecting that around but this approach is not yet
# Line 647  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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