| 924 |
|
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| 925 |
|
|
| 926 |
|
|
| 927 |
\subsection{Crank-Nickelson barotropic time stepping} |
\subsection{Crank-Nicolson barotropic time stepping} |
| 928 |
\label{sec:freesurf-CrankNick} |
\label{sec:freesurf-CrankNick} |
| 929 |
|
|
| 930 |
The full implicit time stepping described previously is |
The full implicit time stepping described previously is |
| 936 |
\\ |
\\ |
| 937 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
| 938 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
| 939 |
stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
stable, Crank-Nicolson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
| 940 |
corresponds to the forward - backward scheme that conserves energy but is |
corresponds to the forward - backward scheme that conserves energy but is |
| 941 |
only stable for small time steps.\\ |
only stable for small time steps.\\ |
| 942 |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
| 999 |
{\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}). |
{\bf useRealFreshWater}{\em=TRUE} in parameter file {\em data}). |
| 1000 |
In order to remain consistent with the tracer equation, specially in |
In order to remain consistent with the tracer equation, specially in |
| 1001 |
the non-linear free-surface formulation, this term is also |
the non-linear free-surface formulation, this term is also |
| 1002 |
affected by the Crank-Nickelson time stepping. The RHS reads: |
affected by the Crank-Nicolson time stepping. The RHS reads: |
| 1003 |
$\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$ |
$\epsilon_{fw} ( \gamma (P-E)^{n+1/2} + (1-\gamma) (P-E)^{n-1/2} )$ |
| 1004 |
%\item The non-hydrostatic part of the code has not yet been |
%\item The non-hydrostatic part of the code has not yet been |
| 1005 |
%updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$. |
%updated, and therefore cannot be used with $(\beta,\gamma) \neq (1,1)$. |
| 1006 |
\item The stability criteria with Crank-Nickelson time stepping |
\item The stability criteria with Crank-Nicolson time stepping |
| 1007 |
for the pure linear gravity wave problem in cartesian coordinates is: |
for the pure linear gravity wave problem in cartesian coordinates is: |
| 1008 |
\begin{itemize} |
\begin{itemize} |
| 1009 |
\item $\beta + \gamma < 1$ : unstable |
\item $\beta + \gamma < 1$ : unstable |
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