/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.20 by jmc, Sun Oct 17 04:14:21 2004 UTC revision 1.24 by edhill, Tue Jun 27 22:31:23 2006 UTC
# Line 10  describe the spatial discretization. The Line 10  describe the spatial discretization. The
10  terms are described first, afterwards the schemes that apply to  terms are described first, afterwards the schemes that apply to
11  passive and dynamically active tracers are described.  passive and dynamically active tracers are described.
12    
13    \input{part2/notation}
14    
15  \section{Time-stepping}  \section{Time-stepping}
16  \begin{rawhtml}  \begin{rawhtml}
# Line 608  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 691  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 814  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) Line 815  $\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h)
815    
816    
817  Once ${\eta}^{n+1}$ has been found, substituting into  Once ${\eta}^{n+1}$ has been found, substituting into
818  \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 $\vec{\bf v}^{n+1}$
819  hydrostatic ($\epsilon_{nh}=0$):  if the model is hydrostatic ($\epsilon_{nh}=0$):
820  $$  $$
821  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}  \vec{\bf v}^{n+1} = \vec{\bf v}^{*}
822  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}  - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1}
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