/[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 1.25 by jmc, Wed Jun 28 16:55:53 2006 UTC
# Line 609  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil Line 609  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
609  (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}  (\theta^*,S^*) & = & (\theta^{n},S^{n}) + \Delta t G_{\theta,S}^{(n+1/2)}
610  \label{eq:tstar-staggered} \\  \label{eq:tstar-staggered} \\
611  (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)  (\theta^{n+1},S^{n+1}) & = & {\cal L}^{-1}_{\theta,S} (\theta^*,S^*)
612  \label{eq:t-n+1-staggered} \\  \label{eq:t-n+1-staggered}
613  \end{eqnarray}  \end{eqnarray}
614  The corresponding calling tree is given in  The corresponding calling tree is given in
615  \ref{fig:call-tree-adams-bashforth-staggered}.  \ref{fig:call-tree-adams-bashforth-staggered}.
# Line 692  The momentum equations are discretized i Line 692  The momentum equations are discretized i
692  \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+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1}
693  & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\  & = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\
694  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}  \frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1}
695  & = & \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}
696  \end{eqnarray}  \end{eqnarray}
697  which must satisfy the discrete-in-time depth integrated continuity,  which must satisfy the discrete-in-time depth integrated continuity,
698  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation  equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation
# Line 796  where Line 796  where
796  \begin{eqnarray}  \begin{eqnarray}
797  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
798  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr
799  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}  \: + \: \epsilon_{fw} \Delta t (P-E)^{n}
800  \label{eq-solve2D_rhs}  \label{eq-solve2D_rhs}
801  \end{eqnarray}  \end{eqnarray}
802    
803  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
804  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})  {\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F})
805    
806  $u^*$: {\bf GuNm1} ({\em DYNVARS.h})  $u^*$: {\bf gU} ({\em DYNVARS.h})
807    
808  $v^*$: {\bf GvNm1} ({\em DYNVARS.h})  $v^*$: {\bf gV} ({\em DYNVARS.h})
809    
810  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)  $\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h)
811    
# Line 857  without any consequence on the solution. Line 857  without any consequence on the solution.
857    
858  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
859    
860  $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h)  $\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em NH\_VARS.h)
861    
862  $u^*$: {\bf GuNm1} ({\em DYNVARS.h})  $u^*$: {\bf gU} ({\em DYNVARS.h})
863    
864  $v^*$: {\bf GvNm1} ({\em DYNVARS.h})  $v^*$: {\bf gV} ({\em DYNVARS.h})
865    
866  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})  $u^{n+1}$: {\bf uVel} ({\em DYNVARS.h})
867    
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