/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.33 by jmc, Thu Apr 18 03:36:16 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    
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