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

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-revision 1.23 by edhill,
Tue Jun 27 22:29:56 2006 UTC
+revision 1.28 by jmc,
Mon Aug 30 23:09:19 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}
+ \label{sec:time_stepping}
  \begin{rawhtml}
  <!-- CMIREDIR:time-stepping: -->
  \end{rawhtml}
-Line 50 
 evaluated explicitly in time. Since the
+Line 51 
 evaluated explicitly in time. Since the
  independent of the particular time-stepping scheme chosen we will
  describe first the over-arching algorithm, known as the pressure
  method, with a rigid-lid model in section
- \ref{sect:pressure-method-rigid-lid}. This algorithm is essentially
+ \ref{sec:pressure-method-rigid-lid}. This algorithm is essentially
  unchanged, apart for some coefficients, when the rigid lid assumption
  is replaced with a linearized implicit free-surface, described in
- section \ref{sect:pressure-method-linear-backward}. These two flavors
+ section \ref{sec:pressure-method-linear-backward}. These two flavors
  of the pressure-method encompass all formulations of the model as it
  exists today. The integration of explicit in time terms is out-lined
- in section \ref{sect:adams-bashforth} and put into the context of the
+ in section \ref{sec:adams-bashforth} and put into the context of the
- overall algorithm in sections \ref{sect:adams-bashforth-sync} and
+ overall algorithm in sections \ref{sec:adams-bashforth-sync} and
- \ref{sect:adams-bashforth-staggered}. Inclusion of non-hydrostatic
+ \ref{sec:adams-bashforth-staggered}. Inclusion of non-hydrostatic
  terms requires applying the pressure method in three dimensions
  instead of two and this algorithm modification is described in section
- \ref{sect:non-hydrostatic}. Finally, the free-surface equation may be
+ \ref{sec:non-hydrostatic}. Finally, the free-surface equation may be
  treated more exactly, including non-linear terms, and this is
- described in section \ref{sect:nonlinear-freesurface}.
+ described in section \ref{sec:nonlinear-freesurface}.
  \section{Pressure method with rigid-lid}
- \label{sect:pressure-method-rigid-lid}
+ \label{sec:pressure-method-rigid-lid}
  \begin{rawhtml}
  <!-- CMIREDIR:pressure_method_rigid_lid: -->
  \end{rawhtml}
  \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
+Line 199 
 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{sec: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}
  \section{Pressure method with implicit linear free-surface}
- \label{sect:pressure-method-linear-backward}
+ \label{sec:pressure-method-linear-backward}
  \begin{rawhtml}
  <!-- CMIREDIR:pressure_method_linear_backward: -->
  \end{rawhtml}
-Line 245 
 pressure method is then replaced by the
+Line 252 
 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{sec: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 286 
 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 290 
 pressure-method.
+Line 298 
 pressure-method.
  \section{Explicit time-stepping: Adams-Bashforth}
- \label{sect:adams-bashforth}
+ \label{sec:adams-bashforth}
  \begin{rawhtml}
  <!-- CMIREDIR:adams_bashforth: -->
  \end{rawhtml}
-Line 300 
 time discretization of the explicit term
+Line 308 
 time discretization of the explicit term
  the quasi-second order Adams-Bashforth method for all explicit terms
  in both the momentum and tracer equations. This is still the default
  mode of operation but it is now possible to use alternate schemes for
- tracers (see section \ref{sect:tracer-advection}).
+ tracers (see section \ref{sec:tracer-advection}).
  \begin{figure}
  \begin{center} \fbox{ \begin{minipage}{4.5in} \begin{tabbing}
-Line 317 
 FORWARD\_STEP \\
+Line 325 
 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 358 
 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 375 
 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{sec:implicit-backward-stepping}
  \begin{rawhtml}
  <!-- CMIREDIR:implicit_time-stepping_backward: -->
  \end{rawhtml}
-Line 408 
 unconditionally stable, no-slip boundary
+Line 445 
 unconditionally stable, no-slip boundary
  implicit and are thus cast as a an explicit drag term.
  \section{Synchronous time-stepping: variables co-located in time}
- \label{sect:adams-bashforth-sync}
+ \label{sec:adams-bashforth-sync}
  \begin{rawhtml}
  <!-- CMIREDIR:adams_bashforth_sync: -->
  \end{rawhtml}
  \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 528 
 time-step. The corresponding calling tre
+Line 565 
 time-step. The corresponding calling tre
  \ref{fig:call-tree-adams-bashforth-sync}.
  \section{Staggered baroclinic time-stepping}
- \label{sect:adams-bashforth-staggered}
+ \label{sec:adams-bashforth-staggered}
  \begin{rawhtml}
  <!-- CMIREDIR:adams_bashforth_staggered: -->
  \end{rawhtml}
  \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 673 
 time-level variables and terms correspon
+Line 710 
 time-level variables and terms correspon
  \section{Non-hydrostatic formulation}
- \label{sect:non-hydrostatic}
+ \label{sec:non-hydrostatic}
  \begin{rawhtml}
  <!-- CMIREDIR:non-hydrostatic_formulation: -->
  \end{rawhtml}
-Line 692 
 The momentum equations are discretized i
+Line 729 
 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 765 
 u^{**} & = & u^{*} - \Delta t g \partial
+Line 802 
 u^{**} & = & u^{*} - \Delta t g \partial
  v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\
  \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +
  \partial_{rr} \phi_{nh}^{n+1} & = &
- \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\
+ \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}  \label{eq:phi-nh}\\
  u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\
  v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\
  \partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1}
-Line 776 
 $w^{n+1}$.
+Line 813 
 $w^{n+1}$.
  \section{Variants on the Free Surface}
- \label{sect:free-surface}
+ \label{sec:free-surface}
  We now describe the various formulations of the free-surface that
  include non-linear forms, implicit in time using Crank-Nicholson,
-Line 796 
 where
+Line 833 
 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 894 
 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 889 
 at the same point in the code.
+Line 926 
 at the same point in the code.
  \subsection{Crank-Nickelson barotropic time stepping}
- \label{sect:freesurf-CrankNick}
+ \label{sec:freesurf-CrankNick}
  The full implicit time stepping described previously is
  unconditionally stable but damps the fast gravity waves, resulting in

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