/[MITgcm]/manual/s_algorithm/text/time_stepping.tex

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-revision 1.16 by edhill,
Thu Aug  7 18:27:52 2003 UTC
+revision 1.17 by jmc,
Wed Oct 13 18:50:54 2004 UTC
 Line 87 
 aaa \= aaa \= aaa \= aaa \= aaa \= aaa \
  \> SOLVE\_FOR\_PRESSURE \\
  \>\> CALC\_DIV\_GHAT \` $H\widehat{u^*}$,$H\widehat{v^*}$ (\ref{eq:elliptic}) \\
  \>\> CG2D \` $\eta^{n+1}$ (\ref{eq:elliptic}) \\
- \> THE\_CORRECTION\_STEP  \\
+ \> MOMENTUM\_CORRECTION\_STEP  \\
  \>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\
  \>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid})
  \end{tabbing} \end{minipage} } \end{center}
 Line 294 
 FORWARD\_STEP \\
  \> THERMODYNAMICS \\
  \>\> CALC\_GT \\
  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ \\
- \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
+ \>either\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:adams-bashforth2}) \\
+ \>or\>\> EXTERNAL\_FORCING \` $G_\theta^{(n+1/2)} = G_\theta^{(n+1/2)} + {\cal Q}$ \\
  \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:taustar}) \\
  \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:tau-n+1-implicit})
  \end{tabbing} \end{minipage} } \end{center}
-Line 332 
 simpler to include these terms and this
+Line 333 
 simpler to include these terms and this
  and forcing evolves smoothly. Problems can, and do, arise when forcing
  or motions are high frequency and this corresponds to a reduced
  stability compared to a simple forward time-stepping of such terms.
+ The model offers the possibility to leave the forcing term outside the
+ Adams-Bashforth extrapolation, by turning off the logical flag
+ {\bf forcing\_In\_AB } (parameter file {\em data}, namelist {\em PARM01},
+ default value = True).
  A stability analysis for an oscillation equation should be given at this point.
  \marginpar{AJA needs to find his notes on this...}
-Line 343 
 A stability analysis for a relaxation eq
+Line 348 
 A stability analysis for a relaxation eq
  \section{Implicit time-stepping: backward method}
  Vertical diffusion and viscosity can be treated implicitly in time
- using the backward method which is an intrinsic scheme. For tracers,
+ using the backward method which is an intrinsic scheme.
+ Recently, the option to treat the vertical advection
+ implicitly has been added, but not yet tested; therefore,
+ the description hereafter is limited to diffusion and viscosity.
+ For tracers,
  the time discretized equation is:
  \begin{equation}
  \tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} =
-Line 374 
 Fig.~\ref{fig:call-tree-adams-bashforth}
+Line 383 
 Fig.~\ref{fig:call-tree-adams-bashforth}
  stepping forward a tracer variable such as temperature.
  In order to fit within the pressure method, the implicit viscosity
- must not alter the barotropic flow. In other words, it can on ly
+ must not alter the barotropic flow. In other words, it can only
  redistribute momentum in the vertical. The upshot of this is that
  although vertical viscosity may be backward implicit and
  unconditionally stable, no-slip boundary conditions may not be made
-Line 402 
 is solved to yield the state variables a
+Line 411 
 is solved to yield the state variables a
  \end{figure}
  \begin{figure}
- \begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing}
+ \begin{center} \fbox{ \begin{minipage}{4.7in} \begin{tabbing}
  aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill
  FORWARD\_STEP \\
+ \>\> EXTERNAL\_FIELDS\_LOAD\\
+ \>\> DO\_ATMOSPHERIC\_PHYS \\
+ \>\> DO\_OCEANIC\_PHYS \\
  \> THERMODYNAMICS \\
  \>\> CALC\_GT \\
  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\
-Line 417 
 FORWARD\_STEP \\
+Line 429 
 FORWARD\_STEP \\
  \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\
  \>\> TIMESTEP \` $\vec{\bf v}^*$ (\ref{eq:Gv-n+5-sync}, \ref{eq:vstar-sync}) \\
  \>\> IMPLDIFF \` $\vec{\bf v}^{**}$ (\ref{eq:vstarstar-sync}) \\
+ \> UPDATE\_R\_STAR or UPDATE\_SURF\_DR \` (NonLin-FS only)\\
  \> SOLVE\_FOR\_PRESSURE \\
  \>\> CALC\_DIV\_GHAT \` $\eta^*$ (\ref{eq:nstar-sync}) \\
  \>\> CG2D \` $\eta^{n+1}$ (\ref{eq:elliptic-sync}) \\
- \> THE\_CORRECTION\_STEP  \\
+ \> MOMENTUM\_CORRECTION\_STEP  \\
  \>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\
- \>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:v-n+1-sync})
+ \>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:v-n+1-sync})\\
+ \> TRACERS\_CORRECTION\_STEP  \\
+ \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
+ \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
+ \>\> CONVECTIVE\_ADJUSTMENT \` \\
  \end{tabbing} \end{minipage} } \end{center}
  \caption{
  Calling tree for the overall synchronous algorithm using
- Adams-Bashforth time-stepping.}
+ Adams-Bashforth time-stepping.
+ The place where the model geometry
+ ({\em hFac} factors) is updated is added here but is only relevant
+ for the non-linear free-surface algorithm.
+ For completeness, the external forcing,
+ ocean and atmospheric physics have been added, although they are mainly
+ optional}
  \label{fig:call-tree-adams-bashforth-sync}
  \end{figure}
-Line 455 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
+Line 478 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
  \label{eq:vstar-sync} \\
  \vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )
  \label{eq:vstarstar-sync} \\
- \eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t
+ \eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t
    \nabla \cdot H \widehat{ \vec{\bf v}^{**} }
  \label{eq:nstar-sync} \\
- \nabla \cdot g H \nabla \eta^{n+1} - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
+ \nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
- & = & - \frac{\eta^*}{\Delta t^2}
+ ~ = ~ - \frac{\eta^*}{\Delta t^2}
  \label{eq:elliptic-sync} \\
  \vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1}
  \label{eq:v-n+1-sync}
-Line 478 
 accelerations, stepping forward and solv
+Line 501 
 accelerations, stepping forward and solv
  surface pressure gradient terms, corresponding to equations
  \ref{eq:Gv-n-sync} to \ref{eq:v-n+1-sync}.
  These operations are carried out in subroutines {\em DYNAMCIS}, {\em
- SOLVE\_FOR\_PRESSURE} and {\em THE\_CORRECTION\_STEP}. This, then,
+ SOLVE\_FOR\_PRESSURE} and {\em MOMENTUM\_CORRECTION\_STEP}. This, then,
  represents an entire algorithm for stepping forward the model one
  time-step. The corresponding calling tree is given in
  \ref{fig:call-tree-adams-bashforth-sync}.
-Line 494 
 time-step. The corresponding calling tre
+Line 517 
 time-step. The corresponding calling tre
  A schematic of the explicit Adams-Bashforth and implicit time-stepping
  phases of the algorithm but with staggering in time of thermodynamic
  variables with the flow. Explicit thermodynamics tendencies are
- evaluated at time level $n-1/2$ as a function of the thermodynamics
+ evaluated at time level $n$ as a function of the thermodynamics
- state at that time level $n$ and flow at time $n$ (dotted arrow). The
+ state at that time level $n$ and flow at time $n+1/2$ (dotted arrow). The
- explicit tendency from the previous time level, $n-3/2$, is used to
+ explicit tendency from the previous time level, $n-1$, is used to
- extrapolate tendencies to $n$ (dashed arrow). This extrapolated
+ extrapolate tendencies to $n+1/2$ (dashed arrow). This extrapolated
  tendency allows thermo-dynamics variables to be stably integrated
- forward-in-time to render an estimate ($*$-variables) at the $n+1/2$
+ forward-in-time to render an estimate ($*$-variables) at the $n+1$
  time level (solid arc-arrow). The implicit-in-time operator ${\cal
  L_{\theta,S}}$ is solved to yield the thermodynamic variables at time
- level $n+1/2$. These are then used to calculate the hydrostatic
+ level $n+1$. These are then used to calculate the hydrostatic
  pressure/geo-potential, $\phi_{hyd}$ (vertical arrows). The
- hydrostatic pressure gradient is evaluated directly an time level
+ hydrostatic pressure gradient is evaluated directly at time level
- $n+1/2$ in stepping forward the flow variables from $n$ to $n+1$
+ $n+1$ in stepping forward the flow variables from $n+1/2$ to $n+3/2$
  (solid arc-arrow). }
  \label{fig:adams-bashforth-staggered}
  \end{figure}
-Line 516 
 circumstance, it is more efficient to st
+Line 539 
 circumstance, it is more efficient to st
  thermodynamic variables with the flow
  variables. Fig.~\ref{fig:adams-bashforth-staggered} illustrates the
  staggering and algorithm. The key difference between this and
- Fig.~\ref{fig:adams-bashforth-sync} is that the new thermodynamics
+ Fig.~\ref{fig:adams-bashforth-sync} is that the thermodynamic variables
- fields are used to compute the hydrostatic pressure at time level
+ are solved after the dynamics, using the recently updated flow field.
- $n+1/2$. The essentially allows the gravity wave terms to leap-frog in
+ This essentially allows the gravity wave terms to leap-frog in
  time giving second order accuracy and more stability.
- The essential change in the staggered algorithm is the calculation of
+ The essential change in the staggered algorithm is that the
- hydrostatic pressure which, in the context of the synchronous
+ thermodynamics solver is delayed from half a time step,
- algorithm involves replacing equation \ref{eq:phi-hyd-sync} with
+ allowing the use of the most recent velocities to compute
- \begin{displaymath}
+ the advection terms. Once the thermodynamics fields are
- \phi_{hyd}^n = \int b(\theta^{n+1},S^{n+1}) dr
+ updated, the hydrostatic pressure is computed
- \end{displaymath}
+ to step frowrad the dynamics
- but the pressure gradient must also be taken out of the
+ Note that the pressure gradient must also be taken out of the
  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,
  $n$ and $n+1$, does not give a user the sense of where variables are
  located in time.  Instead, we re-write the entire algorithm,
  \ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the
  position in time of variables appropriately:
  \begin{eqnarray}
- G_{\theta,S}^{n-1/2} & = & G_{\theta,S} ( u^n, \theta^{n-1/2}, S^{n-1/2} )
+ \vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )
- \label{eq:Gt-n-staggered} \\
- G_{\theta,S}^{(n)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n-1/2}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-3/2}
- \label{eq:Gt-n+5-staggered} \\
- (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n)}
- \label{eq:tstar-staggered} \\
- (\theta^{n+1/2},S^{n+1/2}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)
- \label{eq:t-n+1-staggered} \\
- \phi^{n+1/2}_{hyd} & = & \int b(\theta^{n+1/2},S^{n+1/2}) dr
- \label{eq:phi-hyd-staggered} \\
- \vec{\bf G}_{\vec{\bf v}}^{n} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^n )
  \label{eq:Gv-n-staggered} \\
- \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1}
+ \vec{\bf G}_{\vec{\bf v}}^{(n)} & = & (3/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-1/2} - (1/2 + \epsilon_{AB} ) \vec{\bf G}_{\vec{\bf v}}^{n-3/2}
  \label{eq:Gv-n+5-staggered} \\
- \vec{\bf v}^{*} & = & \vec{\bf v}^{n} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n+1/2)} - \nabla \phi_{hyd}^{n+1/2} \right)
+ \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
+ \label{eq:phi-hyd-staggered} \\
+ \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \left( \vec{\bf G}_{\vec{\bf v}}^{(n)} - \nabla \phi_{hyd}^{n} \right)
  \label{eq:vstar-staggered} \\
  \vec{\bf v}^{**} & = & {\cal L}_{\vec{\bf v}}^{-1} ( \vec{\bf v}^* )
  \label{eq:vstarstar-staggered} \\
- \eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t
+ \eta^* & = & \epsilon_{fs} \left( \eta^{n-1/2} + \Delta t (P-E)^n \right)- \Delta t
    \nabla \cdot H \widehat{ \vec{\bf v}^{**} }
  \label{eq:nstar-staggered} \\
- \nabla \cdot g H \nabla \eta^{n+1} - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
+ \nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2}
- & = & - \frac{\eta^*}{\Delta t^2}
+ ~ = ~ - \frac{\eta^*}{\Delta t^2}
  \label{eq:elliptic-staggered} \\
- \vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1}
+ \vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1/2}
- \label{eq:v-n+1-staggered}
+ \label{eq:v-n+1-staggered} \\
+ G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )
+ \label{eq:Gt-n-staggered} \\
+ G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsilon_{AB}) G_{\theta,S}^{n}-(1/2+\epsilon_{AB}) G_{\theta,S}^{n-1}
+ \label{eq:Gt-n+5-staggered} \\
+ (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}
+ \label{eq:tstar-staggered} \\
+ (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)
+ \label{eq:t-n+1-staggered} \\
  \end{eqnarray}
- The calling sequence is unchanged from
+ The corresponding calling tree is given in
- Fig.~\ref{fig:call-tree-adams-bashforth-sync}. The staggered algorithm
+ \ref{fig:call-tree-adams-bashforth-staggered}.
- is activated with the run-time flag {\bf staggerTimeStep=.TRUE.} in
+ The staggered algorithm is activated with the run-time flag
- {\em PARM01} of {\em data}.
+ {\bf staggerTimeStep=.TRUE.} in parameter file {\em data},
+ namelist {\em PARM01}.
+ \begin{figure}
+ \begin{center} \fbox{ \begin{minipage}{4.7in} \begin{tabbing}
+ aaa \= aaa \= aaa \= aaa \= aaa \= aaa \kill
+ FORWARD\_STEP \\
+ \>\> EXTERNAL\_FIELDS\_LOAD\\
+ \>\> DO\_ATMOSPHERIC\_PHYS \\
+ \>\> DO\_OCEANIC\_PHYS \\
+ \> DYNAMICS \\
+ \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-staggered}) \\
+ \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^{n-1/2}$
+     (\ref{eq:Gv-n-staggered})\\
+ \>\> TIMESTEP \` $\vec{\bf v}^*$ (\ref{eq:Gv-n+5-staggered},
+                                   \ref{eq:vstar-staggered}) \\
+ \>\> IMPLDIFF \` $\vec{\bf v}^{**}$ (\ref{eq:vstarstar-staggered}) \\
+ \> UPDATE\_R\_STAR or UPDATE\_SURF\_DR \` (NonLin-FS only)\\
+ \> SOLVE\_FOR\_PRESSURE \\
+ \>\> CALC\_DIV\_GHAT \` $\eta^*$ (\ref{eq:nstar-staggered}) \\
+ \>\> CG2D \` $\eta^{n+1/2}$ (\ref{eq:elliptic-staggered}) \\
+ \> MOMENTUM\_CORRECTION\_STEP  \\
+ \>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1/2}$ \\
+ \>\> CORRECTION\_STEP \` $u^{n+1/2}$,$v^{n+1/2}$ (\ref{eq:v-n+1-staggered})\\
+ \> THERMODYNAMICS \\
+ \>\> CALC\_GT \\
+ \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$
+      (\ref{eq:Gt-n-staggered})\\
+ \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
+ \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-staggered}) \\
+ \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-staggered}) \\
+ \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
+ \> TRACERS\_CORRECTION\_STEP  \\
+ \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
+ \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
+ \>\> CONVECTIVE\_ADJUSTMENT \` \\
+ \end{tabbing} \end{minipage} } \end{center}
+ \caption{
+ Calling tree for the overall staggered algorithm using
+ Adams-Bashforth time-stepping.
+ The place where the model geometry
+ ({\em hFac} factors) is updated is added here but is only relevant
+ for the non-linear free-surface algorithm.
+ }
+ \label{fig:call-tree-adams-bashforth-staggered}
+ \end{figure}
  The only difficulty with this approach is apparent in equation
  \ref{eq:Gt-n-staggered} and illustrated by the dotted arrow
-Line 651 
 the following equations:
+Line 719 
 the following equations:
  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-nh} \\
  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\
  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\
- \eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t
+ \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)
+ & - & \Delta t
    \partial_x H \widehat{u^{*}}
  + \partial_y H \widehat{v^{*}}
  \\
    \partial_x g H \partial_x \eta^{n+1}
  + \partial_y g H \partial_y \eta^{n+1}
- - \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
+ & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
- & = &
+ ~ = ~
  - \frac{\eta^*}{\Delta t^2}
  \label{eq:elliptic-nh}
  \\

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