| 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}. |
| 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 |
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