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

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-revision 1.19 by edhill,
Sat Oct 16 03:40:12 2004 UTC
+revision 1.25 by jmc,
Wed Jun 28 16:55:53 2006 UTC
 Line 10 
 describe the spatial discretization. The
  terms are described first, afterwards the schemes that apply to
  passive and dynamically active tracers are described.
+ \input{part2/notation}
  \section{Time-stepping}
  \begin{rawhtml}
-Line 225 
 The rigid-lid approximation can be easil
+Line 226 
 The rigid-lid approximation can be easil
  of the free-surface equation which can be written:
  \begin{equation}
  \partial_t \eta + \partial_x H \widehat{u} + \partial_y H \widehat{v} = P-E+R
- \label{eq:linear-free-surface=P-E+R}
+ \label{eq:linear-free-surface=P-E}
  \end{equation}
  which differs from the depth integrated continuity equation with
  rigid-lid (\ref{eq:rigid-lid-continuity}) by the time-dependent term
-Line 233 
 and fresh-water source term.
+Line 234 
 and fresh-water source term.
  Equation \ref{eq:discrete-time-cont-rigid-lid} in the rigid-lid
  pressure method is then replaced by the time discretization of
- \ref{eq:linear-free-surface=P-E+R} which is:
+ \ref{eq:linear-free-surface=P-E} which is:
  \begin{equation}
  \eta^{n+1}
  + \Delta t \partial_x H \widehat{u^{n+1}}
  + \Delta t \partial_y H \widehat{v^{n+1}}
  =
  \eta^{n}
- + \Delta t ( P - E + R )
+ + \Delta t ( P - E )
  \label{eq:discrete-time-backward-free-surface}
  \end{equation}
  where the use of flow at time level $n+1$ makes the method implicit
-Line 255 
 As for the rigid-lid pressure method, eq
+Line 256 
 As for the rigid-lid pressure method, eq
  \begin{eqnarray}
  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\
  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\
- \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}
+ & + & \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-backward-free-surface}
  \\
-Line 442 
 FORWARD\_STEP \\
+Line 443 
 FORWARD\_STEP \\
  \>\>\> GAD\_CALC\_RHS \` $G_\theta^n = G_\theta( u, \theta^n )$ (\ref{eq:Gt-n-sync})\\
  \>\>\> EXTERNAL\_FORCING \` $G_\theta^n = G_\theta^n + {\cal Q}$ \\
  \>\>\> ADAMS\_BASHFORTH2 \` $G_\theta^{(n+1/2)}$ (\ref{eq:Gt-n+5-sync}) \\
- \>\> TIMESTEP\_TRACER \` $\tau^*$ (\ref{eq:tstar-sync}) \\
+ \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-sync}) \\
- \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
+ \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-sync}) \\
  \> DYNAMICS \\
  \>\> CALC\_PHI\_HYD \` $\phi_{hyd}^n$ (\ref{eq:phi-hyd-sync}) \\
  \>\> MOM\_FLUXFORM or MOM\_VECINV \` $G_{\vec{\bf v}}^n$ (\ref{eq:Gv-n-sync})\\
-Line 465 
 FORWARD\_STEP \\
+Line 466 
 FORWARD\_STEP \\
  Calling tree for the overall synchronous algorithm using
  Adams-Bashforth time-stepping.
  The place where the model geometry
- ({\em hFac} factors) is updated is added here but is only relevant
+ ({\bf 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
-Line 547 
 extrapolate tendencies to $n$ (dashed ar
+Line 548 
 extrapolate tendencies to $n$ (dashed ar
  The hydrostatic pressure/geo-potential $\phi_{hyd}$ is evaluated directly
  at time level $n$ (vertical arrows) and used with the extrapolated tendencies
  to step forward the flow variables from $n-1/2$ to $n+1/2$ (solid arc-arrow).
- The implicit-in-time operator ${\cal L_{u,v}}$ (vertical arrows) is
+ The implicit-in-time operator ${\cal L}_{\bf u,v}$ (vertical arrows) is
  then applied to the previous estimation of the the flow field ($*$-variables)
  and yields to the two velocity components $u,v$ at time level $n+1/2$.
  These are then used to calculate the advection term (dashed arc-arrow)
-Line 575 
 thermodynamics solver is delayed from ha
+Line 576 
 thermodynamics solver is delayed from ha
  allowing the use of the most recent velocities to compute
  the advection terms. Once the thermodynamics fields are
  updated, the hydrostatic pressure is computed
- to step frowrad the dynamics
+ to step forwrad the dynamics.
  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
-Line 583 
 located in time.  Instead, we re-write t
+Line 584 
 located in time.  Instead, we re-write t
  \ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the
  position in time of variables appropriately:
  \begin{eqnarray}
+ \phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n}) dr
+ \label{eq:phi-hyd-staggered} \\
  \vec{\bf G}_{\vec{\bf v}}^{n-1/2} & = & \vec{\bf G}_{\vec{\bf v}} ( \vec{\bf v}^{n-1/2} )
  \label{eq:Gv-n-staggered} \\
  \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} \\
- \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}^* )
-Line 608 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
+Line 609 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
  (\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} \\
+ \label{eq:t-n+1-staggered}
  \end{eqnarray}
  The corresponding calling tree is given in
  \ref{fig:call-tree-adams-bashforth-staggered}.
  The staggered algorithm is activated with the run-time flag
- {\bf staggerTimeStep=.TRUE.} in parameter file {\em data},
+ {\bf staggerTimeStep}{\em=.TRUE.} in parameter file {\em data},
  namelist {\em PARM01}.
  \begin{figure}
-Line 643 
 FORWARD\_STEP \\
+Line 644 
 FORWARD\_STEP \\
       (\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}) \\
+ \>\> TIMESTEP\_TRACER \` $\theta^*$ (\ref{eq:tstar-staggered}) \\
- \>\> IMPLDIFF \` $\tau^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
+ \>\> IMPLDIFF \` $\theta^{(n+1)}$ (\ref{eq:t-n+1-staggered}) \\
  \> TRACERS\_CORRECTION\_STEP  \\
  \>\> CYCLE\_TRACER \` $\theta^{n+1}$ \\
  \>\> FILTER \` Shapiro Filter, Zonal Filter (FFT) \\
-Line 654 
 FORWARD\_STEP \\
+Line 655 
 FORWARD\_STEP \\
  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
+ ({\bf 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}
-Line 691 
 The momentum equations are discretized i
+Line 692 
 The momentum equations are discretized i
  \frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1}
  & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\
  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}
- & = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} \\
+ & = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh}
  \end{eqnarray}
  which must satisfy the discrete-in-time depth integrated continuity,
  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation
-Line 775 
 $w^{n+1}$.
+Line 776 
 $w^{n+1}$.
  \section{Variants on the Free Surface}
+ \label{sect:free-surface}
  We now describe the various formulations of the free-surface that
  include non-linear forms, implicit in time using Crank-Nicholson,
-Line 794 
 where
+Line 796 
 where
  \begin{eqnarray}
  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr
- \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}
+ \: + \: \epsilon_{fw} \Delta t (P-E)^{n}
  \label{eq-solve2D_rhs}
  \end{eqnarray}
  \fbox{ \begin{minipage}{4.75in}
  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})
- $u^*$: {\bf GuNm1} ({\em DYNVARS.h})
+ $u^*$: {\bf gU} ({\em DYNVARS.h})
- $v^*$: {\bf GvNm1} ({\em DYNVARS.h})
+ $v^*$: {\bf gV} ({\em DYNVARS.h})
  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)
-Line 813 
 $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
+Line 815 
 $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
  Once ${\eta}^{n+1}$ has been found, substituting into
- \ref{eq:discrete-time-u,eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ if the model is
+ \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$
- hydrostatic ($\epsilon_{nh}=0$):
+ if the model is hydrostatic ($\epsilon_{nh}=0$):
  $$
  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
-Line 855 
 without any consequence on the solution.
+Line 857 
 without any consequence on the solution.
  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
- $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h)
+ $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em NH\_VARS.h)
- $u^*$: {\bf GuNm1} ({\em DYNVARS.h})
+ $u^*$: {\bf gU} ({\em DYNVARS.h})
- $v^*$: {\bf GvNm1} ({\em DYNVARS.h})
+ $v^*$: {\bf gV} ({\em DYNVARS.h})
  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})
-Line 887 
 at the same point in the code.
+Line 889 
 at the same point in the code.
  \subsection{Crank-Nickelson barotropic time stepping}
+ \label{sect:freesurf-CrankNick}
  The full implicit time stepping described previously is
  unconditionally stable but damps the fast gravity waves, resulting in
-Line 901 
 stable, Crank-Nickelson scheme; $(\beta,
+Line 904 
 stable, Crank-Nickelson scheme; $(\beta,
  corresponds to the forward - backward scheme that conserves energy but is
  only stable for small time steps.\\
  In the code, $\beta,\gamma$ are defined as parameters, respectively
- {\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from
+ {\bf implicSurfPress}, {\bf implicDiv2DFlow}. They are read from
- the main data file "{\it data}" and are set by default to 1,1.
+ the main parameter file "{\em data}" and are set by default to 1,1.
  Equations \ref{eq:ustar-backward-free-surface} --
  \ref{eq:vn+1-backward-free-surface} are modified as follows:
- $$
+ \begin{eqnarray*}
  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }
  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]
  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}
   = \frac{ \vec{\bf v}^* }{ \Delta t }
- $$
+ \end{eqnarray*}
- $$
+ \begin{eqnarray}
  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr
  = \epsilon_{fw} (P-E)
- $$
+ \label{eq:eta-n+1-CrankNick}
+ \end{eqnarray}
  where:
  \begin{eqnarray*}
  \vec{\bf v}^* & = &
-Line 939 
 $$
+Line 943 
 $$
  {\bf \nabla}_h {\eta}^{n+1}
  = {\eta}^*
  $$
- and then to compute (correction step):
+ and then to compute ({\em CORRECTION\_STEP}):
  $$
  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
  - \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
  $$
- The non-hydrostatic part is solved as described previously.
+ %The non-hydrostatic part is solved as described previously.
- Note that:
+ \noindent
+ Notes:
  \begin{enumerate}
+ \item The RHS term of equation \ref{eq:eta-n+1-CrankNick}
+ corresponds the contribution of fresh water flux (P-E)
+ to the free-surface variations ($\epsilon_{fw}=1$,
+ {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).
+ In order to remain consistent with the tracer equation, specially in
+ the non-linear free-surface formulation, this term is also
+ affected by the Crank-Nickelson time stepping. The RHS reads:
+ $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$
  \item The non-hydrostatic part of the code has not yet been
- updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$.
+ updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.
  \item The stability criteria with Crank-Nickelson time stepping
  for the pure linear gravity wave problem in cartesian coordinates is:
  \begin{itemize}

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