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

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-revision 1.22 by jmc,
Tue Jun 27 19:10:32 2006 UTC
+revision 1.27 by jmc,
Fri Aug 27 13:08:18 2010 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}
+ \input{s_algorithm/text/notation}
  \section{Time-stepping}
  \begin{rawhtml}
 Line 74 
 described in section \ref{sect:nonlinear
  \begin{figure}
  \begin{center}
- \resizebox{4.0in}{!}{\includegraphics{part2/pressure-method-rigid-lid.eps}}
+ \resizebox{4.0in}{!}{\includegraphics{s_algorithm/figs/pressure-method-rigid-lid.eps}}
  \end{center}
  \caption{
  A schematic of the evolution in time of the pressure method
 Line 198 
 The calling tree for these routines is g
  Fig.~\ref{fig:call-tree-pressure-method}.
+ %\paragraph{Need to discuss implicit viscosity somewhere:}
- \paragraph{Need to discuss implicit viscosity somewhere:}
+ In general, the horizontal momentum time-stepping can contain some terms
+ that are treated implicitly in time,
+ such as the vertical viscosity when using the backward time-stepping scheme
+ (\varlink{implicitViscosity}{implicitViscosity} {\it =.TRUE.}).
+ The method used to solve those implicit terms is provided in
+ section \ref{sect:implicit-backward-stepping}, and modifies
+ equations \ref{eq:discrete-time-u} and \ref{eq:discrete-time-v} to
+ give:
  \begin{eqnarray}
- \frac{1}{\Delta t} u^{n+1} - \partial_z A_v \partial_z u^{n+1}
+ u^{n+1} - \Delta t \partial_z A_v \partial_z u^{n+1}
- + g \partial_x \eta^{n+1} & = & \frac{1}{\Delta t} u^{n} +
+ + \Delta t g \partial_x \eta^{n+1} & = & u^{n} + \Delta t G_u^{(n+1/2)}
- G_u^{(n+1/2)}
  \\
- \frac{1}{\Delta t} v^{n+1} - \partial_z A_v \partial_z v^{n+1}
+ v^{n+1} - \Delta t \partial_z A_v \partial_z v^{n+1}
- + g \partial_y \eta^{n+1} & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)}
+ + \Delta t g \partial_y \eta^{n+1} & = & v^{n} + \Delta t G_v^{(n+1/2)}
  \end{eqnarray}
-Line 245 
 pressure method is then replaced by the
+Line 251 
 pressure method is then replaced by the
  \label{eq:discrete-time-backward-free-surface}
  \end{equation}
  where the use of flow at time level $n+1$ makes the method implicit
- and backward in time. The is the preferred scheme since it still
+ and backward in time. This is the preferred scheme since it still
  filters the fast unresolved wave motions by damping them. A centered
- scheme, such as Crank-Nicholson, would alias the energy of the fast
+ scheme, such as Crank-Nicholson (see section \ref{sect:freesurf-CrankNick}),
- modes onto slower modes of motion.
+ would alias the energy of the fast modes onto slower modes of motion.
  As for the rigid-lid pressure method, equations
  \ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and
-Line 279 
 recovered. However, the implicit treatme
+Line 285 
 recovered. However, the implicit treatme
  the flow to be divergent and for the surface pressure/elevation to
  respond on a finite time-scale (as opposed to instantly). To recover
  the rigid-lid formulation, we introduced a switch-like parameter,
- $\epsilon_{fs}$, which selects between the free-surface and rigid-lid;
+ $\epsilon_{fs}$ (\varlink{freesurfFac}{freesurfFac}),
+ which selects between the free-surface and rigid-lid;
  $\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$
  imposes the rigid-lid. The evolution in time and location of variables
  is exactly as it was for the rigid-lid model so that
-Line 317 
 FORWARD\_STEP \\
+Line 324 
 FORWARD\_STEP \\
  \end{tabbing} \end{minipage} } \end{center}
  \caption{
  Calling tree for the Adams-Bashforth time-stepping of temperature with
- implicit diffusion.}
+ implicit diffusion.
+   (\filelink{THERMODYNAMICS}{model-src-thermodynamics.F},
+    \filelink{ADAMS\_BASHFORTH2}{model-src-adams_bashforth2.F})}
  \label{fig:call-tree-adams-bashforth}
  \end{figure}
-Line 348 
 simpler to include these terms and this
+Line 357 
 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
+ The model offers the possibility to leave the tracer and momentum
- Adams-Bashforth extrapolation, by turning off the logical flag
+ forcing terms and the dissipation terms outside the
- {\bf forcing\_In\_AB } (parameter file {\em data}, namelist {\em PARM01},
+ Adams-Bashforth extrapolation, by turning off the logical flags
- default value = True).
+ \varlink{forcing\_In\_AB}{forcing_In_AB}
+ (parameter file {\em data}, namelist {\em PARM01}, default value = True).
+ (\varlink{tracForcingOutAB}{tracForcingOutAB}, default=0,
+ \varlink{momForcingOutAB}{momForcingOutAB}, default=0)
+ and \varlink{momDissip\_In\_AB}{momDissip_In_AB}
+ (parameter file {\em data}, namelist {\em PARM01}, default value = True).
+ respectively.
  A stability analysis for an oscillation equation should be given at this point.
  \marginpar{AJA needs to find his notes on this...}
-Line 359 
 A stability analysis for an oscillation
+Line 374 
 A stability analysis for an oscillation
  A stability analysis for a relaxation equation should be given at this point.
  \marginpar{...and for this too.}
+ \begin{figure}
+ \begin{center}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/oscil+damp_AB2.eps}}
+ \end{center}
+ \caption{
+ Oscillatory and damping response of
+ quasi-second order Adams-Bashforth scheme for different values
+ of the $\epsilon_{AB}$ parameter (0., 0.1, 0.25, from top to bottom)
+ The analytical solution (in black), the physical mode (in blue)
+ and the numerical mode (in red) are represented with a CFL
+ step of 0.1.
+ The left column represents the oscillatory response
+ on the complex plane for CFL ranging from 0.1 up to 0.9.
+ The right column represents the damping response amplitude
+ (y-axis) function of the CFL (x-axis).
+ }
+ \label{fig:adams-bashforth-respons}
+ \end{figure}
  \section{Implicit time-stepping: backward method}
+ \label{sect:implicit-backward-stepping}
  \begin{rawhtml}
  <!-- CMIREDIR:implicit_time-stepping_backward: -->
  \end{rawhtml}
-Line 415 
 implicit and are thus cast as a an expli
+Line 451 
 implicit and are thus cast as a an expli
  \begin{figure}
  \begin{center}
- \resizebox{5.0in}{!}{\includegraphics{part2/adams-bashforth-sync.eps}}
+ \resizebox{5.0in}{!}{\includegraphics{s_algorithm/figs/adams-bashforth-sync.eps}}
  \end{center}
  \caption{
  A schematic of the explicit Adams-Bashforth and implicit time-stepping
-Line 535 
 time-step. The corresponding calling tre
+Line 571 
 time-step. The corresponding calling tre
  \begin{figure}
  \begin{center}
- \resizebox{5.5in}{!}{\includegraphics{part2/adams-bashforth-staggered.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/adams-bashforth-staggered.eps}}
  \end{center}
  \caption{
  A schematic of the explicit Adams-Bashforth and implicit time-stepping
-Line 609 
 G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
+Line 645 
 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}.
-Line 692 
 The momentum equations are discretized i
+Line 728 
 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 796 
 where
+Line 832 
 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 857 
 without any consequence on the solution.
+Line 893 
 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})

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