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% $Header: /u/gcmpack/mitgcmdoc/part2/time_stepping.tex,v 1.2 2001/08/09 16:22:09 jmc Exp $ |
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% $Name: $ |
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|
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The convention used in this section is as follows: |
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Time is "discretize" using a time step $\Delta t$ |
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and $\Phi^n$ refers to the variable $\Phi$ |
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at time $t = n \Delta t$ . We used the notation $\Phi^{(n)}$ |
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when time interpolation is required to estimate the value of $\phi$ |
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at the time $n \Delta t$. |
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|
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\section{Time integration} |
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|
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The discretization in time of the model equations (cf section I ) |
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does not depend of the discretization in space of each |
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term, so that this section can be read independently. |
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Also for this purpose, we will refers to the continuous |
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space-derivative form of model equations, and focus on |
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the time discretization. |
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|
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The continuous form of the model equations is: |
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|
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\begin{eqnarray} |
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\partial_t \theta & = & G_\theta |
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\\ |
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\partial_t S & = & G_s |
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\\ |
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b' & = & b'(\theta,S,r) |
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\\ |
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\partial_r \phi'_{hyd} & = & -b' |
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\label{eq-r-split-hyd} |
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\\ |
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\partial_t \vec{\bf v} |
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+ {\bf \nabla}_h b_s \eta |
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+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
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& = & \vec{\bf G}_{\vec{\bf v}} |
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- {\bf \nabla}_h \phi'_{hyd} |
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\label{eq-r-split-hmom} |
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\\ |
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\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
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+ \epsilon_{nh} \partial_r \phi'_{nh} |
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& = & \epsilon_{nh} G_{\dot{r}} |
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\label{eq-r-split-rdot} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
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& = & 0 |
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\label{eq-r-cont} |
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\end{eqnarray} |
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where |
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\begin{eqnarray*} |
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G_\theta & = & |
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- \vec{\bf v} \cdot {\bf \nabla} \theta + {\cal Q}_\theta |
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\\ |
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G_S & = & |
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- \vec{\bf v} \cdot {\bf \nabla} S + {\cal Q}_S |
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\\ |
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\vec{\bf G}_{\vec{\bf v}} |
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& = & |
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- \vec{\bf v} \cdot {\bf \nabla} \vec{\bf v} |
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- f \hat{\bf k} \wedge \vec{\bf v} |
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+ \vec{\cal F}_{\vec{\bf v}} |
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\\ |
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G_{\dot{r}} |
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& = & |
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- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
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+ {\cal F}_{\dot{r}} |
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\end{eqnarray*} |
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The exact form of all the "{\it G}"s terms is described in the next |
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section (). Here its sufficient to mention that they contains |
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all the RHS terms except the pressure / geo- potential terms. |
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|
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The switch $\epsilon_{nh}$ allows to activate the non hydrostatic |
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mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, |
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in the hydrostatic limit $\epsilon_{nh} = 0$ |
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and equation \ref{eq-r-split-rdot} vanishes. |
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|
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The equation for $\eta$ is obtained by integrating the |
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continuity equation over the entire depth of the fluid, |
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from $R_{min}(x,y)$ up to $R_o(x,y)$ |
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(Linear free surface): |
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\begin{eqnarray} |
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\epsilon_{fs} \partial_t \eta = |
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\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
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- {\bf \nabla} \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr |
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+ \epsilon_{fw} (P-E) |
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\label{eq-cont-2D} |
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\end{eqnarray} |
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|
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Where $\epsilon_{fs}$,$\epsilon_{fw}$ are two flags to |
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distinguish between a free-surface equation ($\epsilon_{fs}=1$) |
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or the rigid-lid approximation ($\epsilon_{fs}=0$); |
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and to distinguish when exchange of Fresh-Water is included |
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at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) |
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or not ($\epsilon_{fw} = 0$). |
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|
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The hydrostatic potential is found by |
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integrating \ref{eq-r-split-hyd} with the boundary condition that |
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$\phi'_{hyd}(r=R_o) = 0$: |
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\begin{eqnarray*} |
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& & |
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\int_{r'}^{R_o} \partial_r \phi'_{hyd} dr = |
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\left[ \phi'_{hyd} \right]_{r'}^{R_o} = |
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\int_{r'}^{R_o} - b' dr |
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\\ |
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\Rightarrow & & |
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\phi'_{hyd}(x,y,r') = \int_{r'}^{R_o} b' dr |
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\end{eqnarray*} |
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|
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\subsection{General method} |
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|
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The general algorithm consist in a "predictor step" that computes |
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the forward tendencies ("G" terms") and all |
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the "first guess" values star notation): |
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$\vec{\bf v}^*, \theta^*, S^*$ (and $\dot{r}^*$ |
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in non-hydrostatic mode). This is done in the routine {\it DYNAMICS}. |
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|
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Then the implicit terms that appear here on the left hand side (LHS), |
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are solved as follows: |
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Since implicit vertical diffusion and viscosity terms |
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are independent from the barotropic flow adjustment, |
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they are computed first, solving a 3 diagonal Nr x Nr linear system, |
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and incorporated at the end of the {\it DYNAMICS} routines. |
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Then the surface pressure and non hydrostatic pressure |
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are evaluated ({\it SOLVE\_FOR\_PRESSURE}); |
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|
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Finally, within a "corrector step', |
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(routine {\it THE\_CORRECTION\_STEP}) |
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the new values of $u,v,w,\theta,S$ |
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are derived according to the above equations |
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(see details in II.1.3). |
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|
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At this point, the regular time step is over, but |
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the correction step contains also other optional |
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adjustments such as convective adjustment algorithm, or filters |
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(zonal FFT filter, shapiro filter) |
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that applied on both momentum and tracer fields. |
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just before setting the $n+1$ new time step value. |
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|
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Since the pressure solver precision is of the order of |
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the "target residual" that could be lower than the |
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the computer truncation error, and also because some filters |
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might alter the divergence part of the flow field, |
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a final evaluation of the surface r anomaly $\eta^{n+1}$ |
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is performed, according to \ref{eq-rtd-eta} ({\it CALC\_EXACT\_ETA}). |
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This ensures a perfect volume conservation. |
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Note that there is no need for an equivalent Non-hydrostatic |
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"exact conservation" step, since W is already computed after |
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the filters applied. |
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|
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optionnal forcing terms (package):\\ |
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Regarding optional forcing terms (usually part of a "package"), |
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that a account for a specific source or sink term (e.g.: condensation |
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as a sink of water vapor Q), they are generally incorporated |
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in the main algorithm as follows; |
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At the the beginning of the time step, |
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the additionnal tendencies are computed |
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as function of the present state (time step $n$) and external forcing ; |
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Then within the main part of model, |
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only those new tendencies are added to the model variables. |
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|
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[more details needed]\\ |
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The atmospheric physics follows this general scheme. |
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|
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\subsection{Standard synchronous time stepping} |
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|
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In the standard formulation, the surface pressure is |
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evaluated at time step n+1 (implicit method). |
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The above set of equations is then discretized in time |
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as follows: |
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\begin{eqnarray} |
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\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
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\theta^{n+1} & = & \theta^* |
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\\ |
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\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
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S^{n+1} & = & S^* |
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\\ |
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%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
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%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
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%\\ |
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{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
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\label{eq-rtd-hyd} |
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\\ |
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\vec{\bf v} ^{n+1} |
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+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
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+ \epsilon_{nh} \Delta t {\bf \nabla} {\phi'_{nh}}^{n+1} |
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- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
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& = & |
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\vec{\bf v}^* |
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\label{eq-rtd-hmom} |
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\\ |
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\epsilon_{fs} {\eta}^{n+1} + \Delta t |
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{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
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& = & |
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\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
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\nonumber |
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\\ |
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% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
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\label{eq-rtd-eta} |
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\\ |
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\epsilon_{nh} \left( \dot{r} ^{n+1} |
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+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
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\right) |
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& = & \epsilon_{nh} \dot{r}^* |
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\label{eq-rtd-rdot} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
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& = & 0 |
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\label{eq-rtd-cont} |
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\end{eqnarray} |
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where |
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\begin{eqnarray} |
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\theta^* & = & |
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\theta ^{n} + \Delta t G_{\theta} ^{(n+1/2)} |
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\\ |
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S^* & = & |
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S ^{n} + \Delta t G_{S} ^{(n+1/2)} |
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\\ |
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\vec{\bf v}^* & = & |
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\vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
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+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
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\\ |
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\dot{r}^* & = & |
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\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
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\end{eqnarray} |
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|
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Note that implicit vertical terms (viscosity and diffusivity) are |
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not considered as part of the "{\it G}" terms, but are |
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written separately here. |
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|
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To ensure a second order time discretization for both |
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momentum and tracer, |
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The "G" terms are "extrapolated" forward in time |
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(Adams Bashforth time stepping) |
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from the values computed at time step $n$ and $n-1$ |
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to the time $n+1/2$: |
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$$G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1})$$ |
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A small number for the parameter $\epsilon_{AB}$ is generally used |
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to stabilize this time stepping. |
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|
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In the standard non-stagger formulation, |
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the Adams-Bashforth time stepping is also applied to the |
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hydrostatic (pressure / geo-) potential term $\nabla_h \Phi'_{hyd}$. |
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Note that presently, this term is in fact incorporated to the |
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$\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf gU,gV}). |
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|
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\subsection{Stagger baroclinic time stepping} |
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|
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An alternative is to evaluate $\phi'_{hyd}$ with the |
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new tracer fields, and step forward the momentum after. |
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This option is known as stagger baroclinic time stepping, |
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since tracer and momentum are step forward in time one after the other. |
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It can be activated turning on a running flag parameter |
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{\bf staggerTimeStep} in file "{\it data}"). |
| 253 |
|
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The main advantage of this time stepping compared to a synchronous one, |
| 255 |
is a better stability, specially regarding internal gravity waves, |
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and a very natural implementation of a 2nd order in time |
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hydrostatic pressure / geo- potential term. |
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In the other hand, a synchronous time step might be better |
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for convection problems; Its also make simpler time dependent forcing |
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and diagnostic implementation ; and allows a more efficient threading. |
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|
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Although the stagger time step does not affect deeply the |
| 263 |
structure of the code --- a switch allows to evaluate the |
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hydrostatic pressure / geo- potential from new $\theta,S$ |
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instead of the Adams-Bashforth estimation --- |
| 266 |
this affect the way the time discretization is presented : |
| 267 |
|
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\begin{eqnarray*} |
| 269 |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
| 270 |
\theta^{n+1/2} & = & \theta^* |
| 271 |
\\ |
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\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
| 273 |
S^{n+1/2} & = & S^* |
| 274 |
\end{eqnarray*} |
| 275 |
with |
| 276 |
\begin{eqnarray*} |
| 277 |
\theta^* & = & |
| 278 |
\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
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\\ |
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S^* & = & |
| 281 |
S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
| 282 |
\end{eqnarray*} |
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And |
| 284 |
\begin{eqnarray*} |
| 285 |
%{b'}^{n+1/2} & = & b'(\theta^{n+1/2},S^{n+1/2},r) |
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%\\ |
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%\partial_r {\phi'_{hyd}}^{n+1/2} & = & {-b'}^{n+1/2} |
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{\phi'_{hyd}}^{n+1/2} & = & \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) dr |
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%\label{eq-rtd-hyd} |
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\\ |
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\vec{\bf v} ^{n+1} |
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+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
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+ \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 294 |
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
| 295 |
& = & |
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\vec{\bf v}^* |
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%\label{eq-rtd-hmom} |
| 298 |
\\ |
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\epsilon_{fs} {\eta}^{n+1} + \Delta t |
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{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
| 301 |
& = & |
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\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
| 303 |
\\ |
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\epsilon_{nh} \left( \dot{r} ^{n+1} |
| 305 |
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
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\right) |
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& = & \epsilon_{nh} \dot{r}^* |
| 308 |
%\label{eq-rtd-rdot} |
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\\ |
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{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
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& = & 0 |
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%\label{eq-rtd-cont} |
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\end{eqnarray*} |
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with |
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\begin{eqnarray*} |
| 316 |
\vec{\bf v}^* & = & |
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\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 318 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{n+1/2} |
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\\ |
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\dot{r}^* & = & |
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\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
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\end{eqnarray*} |
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|
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%--------------------------------------------------------------------- |
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|
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\subsection{Surface pressure} |
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|
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Substituting \ref{eq-rtd-hmom} into \ref{eq-rtd-cont}, assuming |
| 329 |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
| 330 |
\begin{eqnarray} |
| 331 |
\epsilon_{fs} {\eta}^{n+1} - |
| 332 |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min}) |
| 333 |
{\bf \nabla}_h b_s {\eta}^{n+1} |
| 334 |
= {\eta}^* |
| 335 |
\label{solve_2d} |
| 336 |
\end{eqnarray} |
| 337 |
where |
| 338 |
\begin{eqnarray} |
| 339 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
| 340 |
\Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr |
| 341 |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
| 342 |
\label{solve_2d_rhs} |
| 343 |
\end{eqnarray} |
| 344 |
|
| 345 |
Once ${\eta}^{n+1}$ has been found substituting into \ref{eq-rtd-hmom} |
| 346 |
would yield $\vec{\bf v}^{n+1}$ if the model is hydrostatic |
| 347 |
($\epsilon_{nh}=0$): |
| 348 |
$$ |
| 349 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 350 |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 351 |
$$ |
| 352 |
|
| 353 |
This is known as the correction step. However, when the model is |
| 354 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 355 |
additional equation for $\phi'_{nh}$. This is obtained by |
| 356 |
substituting \ref{eq-rtd-hmom} and \ref{eq-rtd-rdot} into |
| 357 |
\ref{eq-rtd-cont}: |
| 358 |
\begin{equation} |
| 359 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 360 |
= \frac{1}{\Delta t} \left( |
| 361 |
{\bf \nabla}_h \cdot \vec{\bf v}^{**} + \partial_r \dot{r}^* \right) |
| 362 |
\end{equation} |
| 363 |
where |
| 364 |
\begin{displaymath} |
| 365 |
\vec{\bf v}^{**} = \vec{\bf v}^* - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 366 |
\end{displaymath} |
| 367 |
Note that $\eta^{n+1}$ is also used to update the second RHS term |
| 368 |
$\partial_r \dot{r}^* $ since |
| 369 |
the vertical velocity at the surface ($\dot{r}_{surf}$) |
| 370 |
is evaluated as $(\eta^{n+1} - \eta^n) / \Delta t$. |
| 371 |
|
| 372 |
Finally, the horizontal velocities at the new time level are found by: |
| 373 |
\begin{equation} |
| 374 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{**} |
| 375 |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 376 |
\end{equation} |
| 377 |
and the vertical velocity is found by integrating the continuity |
| 378 |
equation vertically. |
| 379 |
Note that, for convenience regarding the restart procedure, |
| 380 |
the integration of the continuity equation has been |
| 381 |
moved at the beginning of the time step (instead of at the end), |
| 382 |
without any consequence on the solution. |
| 383 |
|
| 384 |
Regarding the implementation, all those computation are done |
| 385 |
within the routine {\it SOLVE\_FOR\_PRESSURE} and its dependent |
| 386 |
{\it CALL}s. |
| 387 |
The standard method to solve the 2D elliptic problem (\ref{solve_2d}) |
| 388 |
uses the conjugate gradient method (routine {\it CG2D}); The |
| 389 |
the solver matrix and conjugate gradient operator are only function |
| 390 |
of the discretized domain and are therefore evaluated separately, |
| 391 |
before the time iteration loop, within {\it INI\_CG2D}. |
| 392 |
The computation of the RHS $\eta^*$ is partly |
| 393 |
done in {\it CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
| 394 |
|
| 395 |
The same method is applied for the non hydrostatic part, using |
| 396 |
a conjugate gradient 3D solver ({\it CG3D}) that is initialized |
| 397 |
in {\it INI\_CG3D}. The RHS terms of 2D and 3D problems |
| 398 |
are computed together, within the same part of the code. |
| 399 |
|
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\newpage |
| 401 |
%----------------------------------------------------------------------------------- |
| 402 |
\subsection{Crank-Nickelson barotropic time stepping} |
| 403 |
|
| 404 |
The full implicit time stepping described previously is unconditionally stable |
| 405 |
but damps the fast gravity waves, resulting in a loss of |
| 406 |
gravity potential energy. |
| 407 |
The modification presented hereafter allows to combine an implicit part |
| 408 |
($\beta,\gamma$) and an explicit part ($1-\beta,1-\gamma$) for the surface |
| 409 |
pressure gradient ($\beta$) and for the barotropic flow divergence ($\gamma$). |
| 410 |
\\ |
| 411 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
| 412 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
| 413 |
stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
| 414 |
corresponds to the forward - backward scheme that conserves energy but is |
| 415 |
only stable for small time steps.\\ |
| 416 |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
| 417 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 418 |
the main data file "{\it data}" and are set by default to 1,1. |
| 419 |
|
| 420 |
Equations \ref{eq-rtd-hmom} and \ref{eq-rtd-eta} are modified as follows: |
| 421 |
$$ |
| 422 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 423 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 424 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 425 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
| 426 |
$$ |
| 427 |
$$ |
| 428 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 429 |
+ {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
| 430 |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
| 431 |
= \epsilon_{fw} (P-E) |
| 432 |
$$ |
| 433 |
where: |
| 434 |
\begin{eqnarray*} |
| 435 |
\vec{\bf v}^* & = & |
| 436 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 437 |
+ (\beta-1) \Delta t {\bf \nabla}_h b_s {\eta}^{n} |
| 438 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 439 |
\\ |
| 440 |
{\eta}^* & = & |
| 441 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
| 442 |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
| 443 |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
| 444 |
\end{eqnarray*} |
| 445 |
\\ |
| 446 |
In the hydrostatic case ($\epsilon_{nh}=0$), |
| 447 |
this allow to find ${\eta}^{n+1}$, according to: |
| 448 |
$$ |
| 449 |
\epsilon_{fs} {\eta}^{n+1} - |
| 450 |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{min}) |
| 451 |
{\bf \nabla}_h {\eta}^{n+1} |
| 452 |
= {\eta}^* |
| 453 |
$$ |
| 454 |
and then to compute (correction step): |
| 455 |
$$ |
| 456 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 457 |
- \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 458 |
$$ |
| 459 |
|
| 460 |
The non-hydrostatic part is solved as described previously. |
| 461 |
\\ \\ |
| 462 |
N.B: |
| 463 |
\\ |
| 464 |
a) The non-hydrostatic part of the code has not yet been |
| 465 |
updated, %since it falls out of the purpose of this test, |
| 466 |
so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$. |
| 467 |
\\ |
| 468 |
b) To remind, the stability criteria with the Crank-Nickelson time stepping |
| 469 |
for the pure linear gravity wave problem in cartesian coordinate is: |
| 470 |
\\ |
| 471 |
$\star$~ $\beta + \gamma < 1$ : unstable |
| 472 |
\\ |
| 473 |
$\star$~ $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
| 474 |
\\ |
| 475 |
$\star$~ $\beta + \gamma \geq 1$ : stable if |
| 476 |
%, for all wave length $(k\Delta x,l\Delta y)$ |
| 477 |
$$ |
| 478 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
| 479 |
$$ |
| 480 |
$$ |
| 481 |
\mbox{with }~ |
| 482 |
%c^2 = 2 g H {\Delta t}^2 |
| 483 |
%(\frac{1-cos 2 \pi / k}{\Delta x^2} |
| 484 |
%+\frac{1-cos 2 \pi / l}{\Delta y^2}) |
| 485 |
%$$ |
| 486 |
%Practically, the most stringent condition is obtained with $k=l=2$ : |
| 487 |
%$$ |
| 488 |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
| 489 |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
| 490 |
$$ |
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