/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.32 by jmc, Tue Feb 26 21:23:56 2013 UTC revision 1.34 by jmc, Mon Sep 25 18:11:06 2017 UTC
# Line 776  integrated, and this constraint is used Line 776  integrated, and this constraint is used
776  equations for $\phi_{nh}^{n+1}$:  equations for $\phi_{nh}^{n+1}$:
777  \begin{equation}  \begin{equation}
778  \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +  \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +
779  \partial_{rr} \phi_{nh}^{n+1} =  \partial_{rr} \phi_{nh}^{n+1} = \left(
780  \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}  \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}
781    \right) / \Delta t
782  \end{equation}  \end{equation}
783    
784  The entire algorithm can be summarized as the sequential solution of  The entire algorithm can be summarized as the sequential solution of
# Line 800  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2 Line 801  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2
801  u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\  u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\
802  v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\  v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\
803  \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +  \partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} +
804  \partial_{rr} \phi_{nh}^{n+1} & = &  \partial_{rr} \phi_{nh}^{n+1} & = & \left(
805  \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}  \label{eq:phi-nh}\\  \partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*}
806    \right) / \Delta t  \label{eq:phi-nh}\\
807  u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\  u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\
808  v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\  v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\
809  \partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1}  \partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1}
# Line 809  v^{n+1} & = & v^{**} - \Delta t \partial Line 811  v^{n+1} & = & v^{**} - \Delta t \partial
811  where the last equation is solved by vertically integrating for  where the last equation is solved by vertically integrating for
812  $w^{n+1}$.  $w^{n+1}$.
813    
   
   
814  \section{Variants on the Free Surface}  \section{Variants on the Free Surface}
815  \label{sec:free-surface}  \label{sec:free-surface}
816    
# Line 924  at the same point in the code. Line 924  at the same point in the code.
924    
925    
926    
927  \subsection{Crank-Nickelson barotropic time stepping}  \subsection{Crank-Nicolson barotropic time stepping}
928  \label{sec:freesurf-CrankNick}  \label{sec:freesurf-CrankNick}
929    
930  The full implicit time stepping described previously is  The full implicit time stepping described previously is
# Line 936  for the barotropic flow divergence ($\ga Line 936  for the barotropic flow divergence ($\ga
936  \\  \\
937  For instance, $\beta=\gamma=1$ is the previous fully implicit scheme;  For instance, $\beta=\gamma=1$ is the previous fully implicit scheme;
938  $\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally  $\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally
939  stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$  stable, Crank-Nicolson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$
940  corresponds to the forward - backward scheme that conserves energy but is  corresponds to the forward - backward scheme that conserves energy but is
941  only stable for small time steps.\\  only stable for small time steps.\\
942  In the code, $\beta,\gamma$ are defined as parameters, respectively  In the code, $\beta,\gamma$ are defined as parameters, respectively
# Line 999  to the free-surface variations ($\epsilo Line 999  to the free-surface variations ($\epsilo
999  {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).  {\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}).
1000  In order to remain consistent with the tracer equation, specially in  In order to remain consistent with the tracer equation, specially in
1001  the non-linear free-surface formulation, this term is also  the non-linear free-surface formulation, this term is also
1002  affected by the Crank-Nickelson time stepping. The RHS reads:  affected by the Crank-Nicolson time stepping. The RHS reads:
1003  $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$  $\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$
1004  %\item The non-hydrostatic part of the code has not yet been  %\item The non-hydrostatic part of the code has not yet been
1005  %updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.  %updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$.
1006  \item The stability criteria with Crank-Nickelson time stepping  \item The stability criteria with Crank-Nicolson time stepping
1007  for the pure linear gravity wave problem in cartesian coordinates is:  for the pure linear gravity wave problem in cartesian coordinates is:
1008  \begin{itemize}  \begin{itemize}
1009  \item $\beta + \gamma < 1$ : unstable  \item $\beta + \gamma < 1$ : unstable
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