/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.28 by jmc, Mon Aug 30 23:09:19 2010 UTC revision 1.31 by jmc, Mon May 9 13:45:05 2011 UTC
# Line 263  As for the rigid-lid pressure method, eq Line 263  As for the rigid-lid pressure method, eq
263  \begin{eqnarray}  \begin{eqnarray}
264  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\
265  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\
266  \eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t  \eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)
267    \partial_x H \widehat{u^{*}}           - \Delta t \left( \partial_x H \widehat{u^{*}}
268  + \partial_y H \widehat{v^{*}}                           + \partial_y H \widehat{v^{*}} \right)
269  \\  \\
270    \partial_x g H \partial_x \eta^{n+1}    \partial_x g H \partial_x \eta^{n+1}
271  & + & \partial_y g H \partial_y \eta^{n+1}  & + & \partial_y g H \partial_y \eta^{n+1}
# Line 425  using the Adams-Bashforth method as desc Line 425  using the Adams-Bashforth method as desc
425  \end{eqnarray}  \end{eqnarray}
426  where ${\cal L}_\tau^{-1}$ is the inverse of the operator  where ${\cal L}_\tau^{-1}$ is the inverse of the operator
427  \begin{equation}  \begin{equation}
428  {\cal L} = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right]  {\cal L}_\tau = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right]
429  \end{equation}  \end{equation}
430  Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar}  Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar}
431  while \ref{eq:tau-n+1-implicit} involves an operator or matrix  while \ref{eq:tau-n+1-implicit} involves an operator or matrix
# Line 542  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil Line 542  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
542  \nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}  \nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
543  ~ = ~ - \frac{\eta^*}{\Delta t^2}  ~ = ~ - \frac{\eta^*}{\Delta t^2}
544  \label{eq:elliptic-sync} \\  \label{eq:elliptic-sync} \\
545  \vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1}  \vec{\bf v}^{n+1} & = & \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1}
546  \label{eq:v-n+1-sync}  \label{eq:v-n+1-sync}
547  \end{eqnarray}  \end{eqnarray}
548  Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of  Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of
# Line 613  thermodynamics solver is delayed from ha Line 613  thermodynamics solver is delayed from ha
613  allowing the use of the most recent velocities to compute  allowing the use of the most recent velocities to compute
614  the advection terms. Once the thermodynamics fields are  the advection terms. Once the thermodynamics fields are
615  updated, the hydrostatic pressure is computed  updated, the hydrostatic pressure is computed
616  to step forwrad the dynamics.  to step forward the dynamics.
617  Note that the pressure gradient must also be taken out of the  Note that the pressure gradient must also be taken out of the
618  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,  Adams-Bashforth extrapolation. Also, retaining the integer time-levels,
619  $n$ and $n+1$, does not give a user the sense of where variables are  $n$ and $n+1$, does not give a user the sense of where variables are
# Line 637  position in time of variables appropriat Line 637  position in time of variables appropriat
637  \nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2}  \nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2}
638  ~ = ~ - \frac{\eta^*}{\Delta t^2}  ~ = ~ - \frac{\eta^*}{\Delta t^2}
639  \label{eq:elliptic-staggered} \\  \label{eq:elliptic-staggered} \\
640  \vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1/2}  \vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1/2}
641  \label{eq:v-n+1-staggered} \\  \label{eq:v-n+1-staggered} \\
642  G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )  G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )
643  \label{eq:Gt-n-staggered} \\  \label{eq:Gt-n-staggered} \\
# Line 718  time-level variables and terms correspon Line 718  time-level variables and terms correspon
718  The non-hydrostatic formulation re-introduces the full vertical  The non-hydrostatic formulation re-introduces the full vertical
719  momentum equation and requires the solution of a 3-D elliptic  momentum equation and requires the solution of a 3-D elliptic
720  equations for non-hydrostatic pressure perturbation. We still  equations for non-hydrostatic pressure perturbation. We still
721  intergrate vertically for the hydrostatic pressure and solve a 2-D  integrate vertically for the hydrostatic pressure and solve a 2-D
722  elliptic equation for the surface pressure/elevation for this reduces  elliptic equation for the surface pressure/elevation for this reduces
723  the amount of work needed to solve for the non-hydrostatic pressure.  the amount of work needed to solve for the non-hydrostatic pressure.
724    
# Line 759  Substituting into the depth integrated c Line 759  Substituting into the depth integrated c
759  \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)  \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)
760  +  +
761  \partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)  \partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)
762   - \frac{\epsilon_{fs}\eta^*}{\Delta t^2}   - \frac{\epsilon_{fs}\eta^{n+1}}{\Delta t^2}
763  = - \frac{\eta^*}{\Delta t^2}  = - \frac{\eta^*}{\Delta t^2}
764  \end{equation}  \end{equation}
765  which is approximated by equation  which is approximated by equation
# Line 767  which is approximated by equation Line 767  which is approximated by equation
767  $\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh}  $\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh}
768  << g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is  << g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is
769  solved accurately then the implication is that $\widehat{\phi}_{nh}  solved accurately then the implication is that $\widehat{\phi}_{nh}
770  \approx 0$ so that thet non-hydrostatic pressure field does not drive  \approx 0$ so that the non-hydrostatic pressure field does not drive
771  barotropic motion.  barotropic motion.
772    
773  The flow must satisfy non-divergence  The flow must satisfy non-divergence
# Line 787  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2 Line 787  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2
787  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\
788  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\
789  \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)  \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)
790  & - & \Delta t  & - & \Delta t \left( \partial_x H \widehat{u^{*}}
791    \partial_x H \widehat{u^{*}}                      + \partial_y H \widehat{v^{*}} \right)
 + \partial_y H \widehat{v^{*}}  
792  \\  \\
793    \partial_x g H \partial_x \eta^{n+1}    \partial_x g H \partial_x \eta^{n+1}
794  + \partial_y g H \partial_y \eta^{n+1}  + \partial_y g H \partial_y \eta^{n+1}
# Line 950  Equations \ref{eq:ustar-backward-free-su Line 949  Equations \ref{eq:ustar-backward-free-su
949  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }  \frac{ \vec{\bf v}^{n+1} }{ \Delta t }
950  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]  + {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ]
951  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}  + \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1}
952   = \frac{ \vec{\bf v}^* }{ \Delta t }   = \frac{ \vec{\bf v}^{n} }{ \Delta t }
953     + \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)}
954     + {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)}
955  \end{eqnarray*}  \end{eqnarray*}
956  \begin{eqnarray}  \begin{eqnarray}
957  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
# Line 959  Equations \ref{eq:ustar-backward-free-su Line 960  Equations \ref{eq:ustar-backward-free-su
960  = \epsilon_{fw} (P-E)  = \epsilon_{fw} (P-E)
961  \label{eq:eta-n+1-CrankNick}  \label{eq:eta-n+1-CrankNick}
962  \end{eqnarray}  \end{eqnarray}
963  where:  We set
964  \begin{eqnarray*}  \begin{eqnarray*}
965  \vec{\bf v}^* & = &  \vec{\bf v}^* & = &
966  \vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)}  \vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)}
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