| 1 |
% $Header: /u/gcmpack/mitgcmdoc/part2/time_stepping.tex,v 1.3 2001/08/09 20:06:18 jmc Exp $ |
| 2 |
% $Name: $ |
| 3 |
|
| 4 |
The convention used in this section is as follows: |
| 5 |
Time is "discretize" using a time step $\Delta t$ |
| 6 |
and $\Phi^n$ refers to the variable $\Phi$ |
| 7 |
at time $t = n \Delta t$ . We used the notation $\Phi^{(n)}$ |
| 8 |
when time interpolation is required to estimate the value of $\phi$ |
| 9 |
at the time $n \Delta t$. |
| 10 |
|
| 11 |
\section{Time integration} |
| 12 |
|
| 13 |
The discretization in time of the model equations (cf section I ) |
| 14 |
does not depend of the discretization in space of each |
| 15 |
term, so that this section can be read independently. |
| 16 |
Also for this purpose, we will refers to the continuous |
| 17 |
space-derivative form of model equations, and focus on |
| 18 |
the time discretization. |
| 19 |
|
| 20 |
The continuous form of the model equations is: |
| 21 |
|
| 22 |
\begin{eqnarray} |
| 23 |
\partial_t \theta & = & G_\theta |
| 24 |
\label{eq-tCsC-theta} |
| 25 |
\\ |
| 26 |
\partial_t S & = & G_s |
| 27 |
\label{eq-tCsC-salt} |
| 28 |
\\ |
| 29 |
b' & = & b'(\theta,S,r) |
| 30 |
\\ |
| 31 |
\partial_r \phi'_{hyd} & = & -b' |
| 32 |
\label{eq-tCsC-hyd} |
| 33 |
\\ |
| 34 |
\partial_t \vec{\bf v} |
| 35 |
+ {\bf \nabla}_h b_s \eta |
| 36 |
+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
| 37 |
& = & \vec{\bf G}_{\vec{\bf v}} |
| 38 |
- {\bf \nabla}_h \phi'_{hyd} |
| 39 |
\label{eq-tCsC-Hmom} |
| 40 |
\\ |
| 41 |
\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
| 42 |
+ \epsilon_{nh} \partial_r \phi'_{nh} |
| 43 |
& = & \epsilon_{nh} G_{\dot{r}} |
| 44 |
\label{eq-tCsC-Vmom} |
| 45 |
\\ |
| 46 |
{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
| 47 |
& = & 0 |
| 48 |
\label{eq-tCsC-cont} |
| 49 |
\end{eqnarray} |
| 50 |
where |
| 51 |
\begin{eqnarray*} |
| 52 |
G_\theta & = & |
| 53 |
- \vec{\bf v} \cdot {\bf \nabla} \theta + {\cal Q}_\theta |
| 54 |
\\ |
| 55 |
G_S & = & |
| 56 |
- \vec{\bf v} \cdot {\bf \nabla} S + {\cal Q}_S |
| 57 |
\\ |
| 58 |
\vec{\bf G}_{\vec{\bf v}} |
| 59 |
& = & |
| 60 |
- \vec{\bf v} \cdot {\bf \nabla} \vec{\bf v} |
| 61 |
- f \hat{\bf k} \wedge \vec{\bf v} |
| 62 |
+ \vec{\cal F}_{\vec{\bf v}} |
| 63 |
\\ |
| 64 |
G_{\dot{r}} |
| 65 |
& = & |
| 66 |
- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
| 67 |
+ {\cal F}_{\dot{r}} |
| 68 |
\end{eqnarray*} |
| 69 |
The exact form of all the "{\it G}"s terms is described in the next |
| 70 |
section (). Here its sufficient to mention that they contains |
| 71 |
all the RHS terms except the pressure / geo- potential terms. |
| 72 |
|
| 73 |
The switch $\epsilon_{nh}$ allows to activate the non hydrostatic |
| 74 |
mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, |
| 75 |
in the hydrostatic limit $\epsilon_{nh} = 0$ |
| 76 |
and equation \ref{eq-tCsC-Vmom} vanishes. |
| 77 |
|
| 78 |
The equation for $\eta$ is obtained by integrating the |
| 79 |
continuity equation over the entire depth of the fluid, |
| 80 |
from $R_{min}(x,y)$ up to $R_o(x,y)$ |
| 81 |
(Linear free surface): |
| 82 |
\begin{eqnarray} |
| 83 |
\epsilon_{fs} \partial_t \eta = |
| 84 |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
| 85 |
- {\bf \nabla} \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr |
| 86 |
+ \epsilon_{fw} (P-E) |
| 87 |
\label{eq-tCsC-eta} |
| 88 |
\end{eqnarray} |
| 89 |
|
| 90 |
Where $\epsilon_{fs}$,$\epsilon_{fw}$ are two flags to |
| 91 |
distinguish between a free-surface equation ($\epsilon_{fs}=1$) |
| 92 |
or the rigid-lid approximation ($\epsilon_{fs}=0$); |
| 93 |
and to distinguish when exchange of Fresh-Water is included |
| 94 |
at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) |
| 95 |
or not ($\epsilon_{fw} = 0$). |
| 96 |
|
| 97 |
The hydrostatic potential is found by |
| 98 |
integrating \ref{eq-tCsC-hyd} with the boundary condition that |
| 99 |
$\phi'_{hyd}(r=R_o) = 0$: |
| 100 |
\begin{eqnarray*} |
| 101 |
& & |
| 102 |
\int_{r'}^{R_o} \partial_r \phi'_{hyd} dr = |
| 103 |
\left[ \phi'_{hyd} \right]_{r'}^{R_o} = |
| 104 |
\int_{r'}^{R_o} - b' dr |
| 105 |
\\ |
| 106 |
\Rightarrow & & |
| 107 |
\phi'_{hyd}(x,y,r') = \int_{r'}^{R_o} b' dr |
| 108 |
\end{eqnarray*} |
| 109 |
|
| 110 |
\subsection{General method} |
| 111 |
|
| 112 |
An overview of the general method is presented hereafter, |
| 113 |
with explicit references to the Fortran code. This part |
| 114 |
often refers to the discretized equations of the model |
| 115 |
that are detailed in the following sections. |
| 116 |
|
| 117 |
The general algorithm consist in a "predictor step" that computes |
| 118 |
the forward tendencies ("G" terms") and all |
| 119 |
the "first guess" values (star notation): |
| 120 |
$\theta^*, S^*, \vec{\bf v}^*$ (and $\dot{r}^*$ |
| 121 |
in non-hydrostatic mode). This is done in the two routines |
| 122 |
{\it THERMODYNAMICS} and {\it DYNAMICS}. |
| 123 |
|
| 124 |
Then the implicit terms that appear on the left hand side (LHS) |
| 125 |
of equations \ref{eq-tDsC-theta} - \ref{eq-tDsC-cont}, |
| 126 |
are solved as follows: |
| 127 |
Since implicit vertical diffusion and viscosity terms |
| 128 |
are independent from the barotropic flow adjustment, |
| 129 |
they are computed first, solving a 3 diagonal Nr x Nr linear system, |
| 130 |
and incorporated at the end of the {\it THERMODYNAMICS} and |
| 131 |
{\it DYNAMICS} routines. |
| 132 |
Then the surface pressure and non hydrostatic pressure |
| 133 |
are evaluated ({\it SOLVE\_FOR\_PRESSURE}); |
| 134 |
|
| 135 |
Finally, within a "corrector step', |
| 136 |
(routine {\it THE\_CORRECTION\_STEP}) |
| 137 |
the new values of $u,v,w,\theta,S$ |
| 138 |
are derived according to the above equations |
| 139 |
(see details in II.1.3). |
| 140 |
|
| 141 |
At this point, the regular time step is over, but |
| 142 |
the correction step contains also other optional |
| 143 |
adjustments such as convective adjustment algorithm, or filters |
| 144 |
(zonal FFT filter, shapiro filter) |
| 145 |
that applied on both momentum and tracer fields. |
| 146 |
just before setting the $n+1$ new time step value. |
| 147 |
|
| 148 |
Since the pressure solver precision is of the order of |
| 149 |
the "target residual" that could be lower than the |
| 150 |
the computer truncation error, and also because some filters |
| 151 |
might alter the divergence part of the flow field, |
| 152 |
a final evaluation of the surface r anomaly $\eta^{n+1}$ |
| 153 |
is performed, according to \ref{eq-tDsC-eta} ({\it CALC\_EXACT\_ETA}). |
| 154 |
This ensures a perfect volume conservation. |
| 155 |
Note that there is no need for an equivalent Non-hydrostatic |
| 156 |
"exact conservation" step, since W is already computed after |
| 157 |
the filters applied. |
| 158 |
|
| 159 |
Regarding optional forcing terms (usually part of a "package"), |
| 160 |
that account for a specific source or sink term (e.g.: condensation |
| 161 |
as a sink of water vapor Q), they are generally incorporated |
| 162 |
in the main algorithm as follows; |
| 163 |
At the the beginning of the time step, |
| 164 |
the additional tendencies are computed |
| 165 |
as function of the present state (time step $n$) and external forcing ; |
| 166 |
Then within the main part of model, |
| 167 |
only those new tendencies are added to the model variables. |
| 168 |
|
| 169 |
[more details needed]\\ |
| 170 |
|
| 171 |
The atmospheric physics follows this general scheme. |
| 172 |
|
| 173 |
[more about C\_grid, A\_grid conversion \& drag term]\\ |
| 174 |
|
| 175 |
\subsection{Standard synchronous time stepping} |
| 176 |
|
| 177 |
In the standard formulation, the surface pressure is |
| 178 |
evaluated at time step n+1 (implicit method). |
| 179 |
The above set of equations is then discretized in time |
| 180 |
as follows: |
| 181 |
\begin{eqnarray} |
| 182 |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
| 183 |
\theta^{n+1} & = & \theta^* |
| 184 |
\label{eq-tDsC-theta} |
| 185 |
\\ |
| 186 |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
| 187 |
S^{n+1} & = & S^* |
| 188 |
\label{eq-tDsC-salt} |
| 189 |
\\ |
| 190 |
%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
| 191 |
%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
| 192 |
%\\ |
| 193 |
{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
| 194 |
\label{eq-tDsC-hyd} |
| 195 |
\\ |
| 196 |
\vec{\bf v} ^{n+1} |
| 197 |
+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 198 |
+ \epsilon_{nh} \Delta t {\bf \nabla} {\phi'_{nh}}^{n+1} |
| 199 |
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
| 200 |
& = & |
| 201 |
\vec{\bf v}^* |
| 202 |
\label{eq-tDsC-Hmom} |
| 203 |
\\ |
| 204 |
\epsilon_{fs} {\eta}^{n+1} + \Delta t |
| 205 |
{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
| 206 |
& = & |
| 207 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
| 208 |
\nonumber |
| 209 |
\\ |
| 210 |
% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
| 211 |
\label{eq-tDsC-eta} |
| 212 |
\\ |
| 213 |
\epsilon_{nh} \left( \dot{r} ^{n+1} |
| 214 |
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
| 215 |
\right) |
| 216 |
& = & \epsilon_{nh} \dot{r}^* |
| 217 |
\label{eq-tDsC-Vmom} |
| 218 |
\\ |
| 219 |
{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
| 220 |
& = & 0 |
| 221 |
\label{eq-tDsC-cont} |
| 222 |
\end{eqnarray} |
| 223 |
where |
| 224 |
\begin{eqnarray} |
| 225 |
\theta^* & = & |
| 226 |
\theta ^{n} + \Delta t G_{\theta} ^{(n+1/2)} |
| 227 |
\\ |
| 228 |
S^* & = & |
| 229 |
S ^{n} + \Delta t G_{S} ^{(n+1/2)} |
| 230 |
\\ |
| 231 |
\vec{\bf v}^* & = & |
| 232 |
\vec{\bf v}^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 233 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 234 |
\\ |
| 235 |
\dot{r}^* & = & |
| 236 |
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
| 237 |
\end{eqnarray} |
| 238 |
|
| 239 |
Note that implicit vertical terms (viscosity and diffusivity) are |
| 240 |
not considered as part of the "{\it G}" terms, but are |
| 241 |
written separately here. |
| 242 |
|
| 243 |
To ensure a second order time discretization for both |
| 244 |
momentum and tracer, |
| 245 |
The "{\it G}" terms are "extrapolated" forward in time |
| 246 |
(Adams Bashforth time stepping) |
| 247 |
from the values computed at time step $n$ and $n-1$ |
| 248 |
to the time $n+1/2$: |
| 249 |
$$G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1})$$ |
| 250 |
A small number for the parameter $\epsilon_{AB}$ is generally used |
| 251 |
to stabilize this time stepping. |
| 252 |
|
| 253 |
In the standard non-stagger formulation, |
| 254 |
the Adams-Bashforth time stepping is also applied to the |
| 255 |
hydrostatic (pressure / geo-) potential term $\nabla_h \Phi'_{hyd}$. |
| 256 |
Note that presently, this term is in fact incorporated to the |
| 257 |
$\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf gU,gV}). |
| 258 |
|
| 259 |
\subsection{Stagger baroclinic time stepping} |
| 260 |
|
| 261 |
An alternative is to evaluate $\phi'_{hyd}$ with the |
| 262 |
new tracer fields, and step forward the momentum after. |
| 263 |
This option is known as stagger baroclinic time stepping, |
| 264 |
since tracer and momentum are step forward in time one after the other. |
| 265 |
It can be activated turning on a running flag parameter |
| 266 |
{\bf staggerTimeStep} in file "{\it data}"). |
| 267 |
|
| 268 |
The main advantage of this time stepping compared to a synchronous one, |
| 269 |
is a better stability, specially regarding internal gravity waves, |
| 270 |
and a very natural implementation of a 2nd order in time |
| 271 |
hydrostatic pressure / geo- potential term. |
| 272 |
In the other hand, a synchronous time step might be better |
| 273 |
for convection problems; Its also make simpler time dependent forcing |
| 274 |
and diagnostic implementation ; and allows a more efficient threading. |
| 275 |
|
| 276 |
Although the stagger time step does not affect deeply the |
| 277 |
structure of the code --- a switch allows to evaluate the |
| 278 |
hydrostatic pressure / geo- potential from new $\theta,S$ |
| 279 |
instead of the Adams-Bashforth estimation --- |
| 280 |
this affect the way the time discretization is presented : |
| 281 |
|
| 282 |
\begin{eqnarray*} |
| 283 |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
| 284 |
\theta^{n+1/2} & = & \theta^* |
| 285 |
\\ |
| 286 |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
| 287 |
S^{n+1/2} & = & S^* |
| 288 |
\end{eqnarray*} |
| 289 |
with |
| 290 |
\begin{eqnarray*} |
| 291 |
\theta^* & = & |
| 292 |
\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
| 293 |
\\ |
| 294 |
S^* & = & |
| 295 |
S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
| 296 |
\end{eqnarray*} |
| 297 |
And |
| 298 |
\begin{eqnarray*} |
| 299 |
%{b'}^{n+1/2} & = & b'(\theta^{n+1/2},S^{n+1/2},r) |
| 300 |
%\\ |
| 301 |
%\partial_r {\phi'_{hyd}}^{n+1/2} & = & {-b'}^{n+1/2} |
| 302 |
{\phi'_{hyd}}^{n+1/2} & = & \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) dr |
| 303 |
%\label{eq-tDsC-hyd} |
| 304 |
\\ |
| 305 |
\vec{\bf v} ^{n+1} |
| 306 |
+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 307 |
+ \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 308 |
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
| 309 |
& = & |
| 310 |
\vec{\bf v}^* |
| 311 |
%\label{eq-tDsC-Hmom} |
| 312 |
\\ |
| 313 |
\epsilon_{fs} {\eta}^{n+1} + \Delta t |
| 314 |
{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
| 315 |
& = & |
| 316 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
| 317 |
\\ |
| 318 |
\epsilon_{nh} \left( \dot{r} ^{n+1} |
| 319 |
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
| 320 |
\right) |
| 321 |
& = & \epsilon_{nh} \dot{r}^* |
| 322 |
%\label{eq-tDsC-Vmom} |
| 323 |
\\ |
| 324 |
{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
| 325 |
& = & 0 |
| 326 |
%\label{eq-tDsC-cont} |
| 327 |
\end{eqnarray*} |
| 328 |
with |
| 329 |
\begin{eqnarray*} |
| 330 |
\vec{\bf v}^* & = & |
| 331 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 332 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{n+1/2} |
| 333 |
\\ |
| 334 |
\dot{r}^* & = & |
| 335 |
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
| 336 |
\end{eqnarray*} |
| 337 |
|
| 338 |
%--------------------------------------------------------------------- |
| 339 |
|
| 340 |
\subsection{Surface pressure} |
| 341 |
|
| 342 |
Substituting \ref{eq-tDsC-Hmom} into \ref{eq-tDsC-cont}, assuming |
| 343 |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
| 344 |
\begin{eqnarray} |
| 345 |
\epsilon_{fs} {\eta}^{n+1} - |
| 346 |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min}) |
| 347 |
{\bf \nabla}_h b_s {\eta}^{n+1} |
| 348 |
= {\eta}^* |
| 349 |
\label{eq-solve2D} |
| 350 |
\end{eqnarray} |
| 351 |
where |
| 352 |
\begin{eqnarray} |
| 353 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
| 354 |
\Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr |
| 355 |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
| 356 |
\label{eq-solve2D_rhs} |
| 357 |
\end{eqnarray} |
| 358 |
|
| 359 |
Once ${\eta}^{n+1}$ has been found substituting into \ref{eq-tDsC-Hmom} |
| 360 |
would yield $\vec{\bf v}^{n+1}$ if the model is hydrostatic |
| 361 |
($\epsilon_{nh}=0$): |
| 362 |
$$ |
| 363 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 364 |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 365 |
$$ |
| 366 |
|
| 367 |
This is known as the correction step. However, when the model is |
| 368 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 369 |
additional equation for $\phi'_{nh}$. This is obtained by |
| 370 |
substituting \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
| 371 |
\ref{eq-tDsC-cont}: |
| 372 |
\begin{equation} |
| 373 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 374 |
= \frac{1}{\Delta t} \left( |
| 375 |
{\bf \nabla}_h \cdot \vec{\bf v}^{**} + \partial_r \dot{r}^* \right) |
| 376 |
\end{equation} |
| 377 |
where |
| 378 |
\begin{displaymath} |
| 379 |
\vec{\bf v}^{**} = \vec{\bf v}^* - \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 380 |
\end{displaymath} |
| 381 |
Note that $\eta^{n+1}$ is also used to update the second RHS term |
| 382 |
$\partial_r \dot{r}^* $ since |
| 383 |
the vertical velocity at the surface ($\dot{r}_{surf}$) |
| 384 |
is evaluated as $(\eta^{n+1} - \eta^n) / \Delta t$. |
| 385 |
|
| 386 |
Finally, the horizontal velocities at the new time level are found by: |
| 387 |
\begin{equation} |
| 388 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{**} |
| 389 |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 390 |
\end{equation} |
| 391 |
and the vertical velocity is found by integrating the continuity |
| 392 |
equation vertically. |
| 393 |
Note that, for convenience regarding the restart procedure, |
| 394 |
the integration of the continuity equation has been |
| 395 |
moved at the beginning of the time step (instead of at the end), |
| 396 |
without any consequence on the solution. |
| 397 |
|
| 398 |
Regarding the implementation, all those computation are done |
| 399 |
within the routine {\it SOLVE\_FOR\_PRESSURE} and its dependent |
| 400 |
{\it CALL}s. |
| 401 |
The standard method to solve the 2D elliptic problem (\ref{eq-solve2D}) |
| 402 |
uses the conjugate gradient method (routine {\it CG2D}); The |
| 403 |
the solver matrix and conjugate gradient operator are only function |
| 404 |
of the discretized domain and are therefore evaluated separately, |
| 405 |
before the time iteration loop, within {\it INI\_CG2D}. |
| 406 |
The computation of the RHS $\eta^*$ is partly |
| 407 |
done in {\it CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
| 408 |
|
| 409 |
The same method is applied for the non hydrostatic part, using |
| 410 |
a conjugate gradient 3D solver ({\it CG3D}) that is initialized |
| 411 |
in {\it INI\_CG3D}. The RHS terms of 2D and 3D problems |
| 412 |
are computed together, within the same part of the code. |
| 413 |
|
| 414 |
\newpage |
| 415 |
%----------------------------------------------------------------------------------- |
| 416 |
\subsection{Crank-Nickelson barotropic time stepping} |
| 417 |
|
| 418 |
The full implicit time stepping described previously is unconditionally stable |
| 419 |
but damps the fast gravity waves, resulting in a loss of |
| 420 |
gravity potential energy. |
| 421 |
The modification presented hereafter allows to combine an implicit part |
| 422 |
($\beta,\gamma$) and an explicit part ($1-\beta,1-\gamma$) for the surface |
| 423 |
pressure gradient ($\beta$) and for the barotropic flow divergence ($\gamma$). |
| 424 |
\\ |
| 425 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
| 426 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
| 427 |
stable, Crank-Nickelson scheme; $(\beta,\gamma)=(1,0)$ or $=(0,1)$ |
| 428 |
corresponds to the forward - backward scheme that conserves energy but is |
| 429 |
only stable for small time steps.\\ |
| 430 |
In the code, $\beta,\gamma$ are defined as parameters, respectively |
| 431 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 432 |
the main data file "{\it data}" and are set by default to 1,1. |
| 433 |
|
| 434 |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
| 435 |
$$ |
| 436 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 437 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 438 |
+ \epsilon_{nh} {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 439 |
= \frac{ \vec{\bf v}^* }{ \Delta t } |
| 440 |
$$ |
| 441 |
$$ |
| 442 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 443 |
+ {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
| 444 |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
| 445 |
= \epsilon_{fw} (P-E) |
| 446 |
$$ |
| 447 |
where: |
| 448 |
\begin{eqnarray*} |
| 449 |
\vec{\bf v}^* & = & |
| 450 |
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
| 451 |
+ (\beta-1) \Delta t {\bf \nabla}_h b_s {\eta}^{n} |
| 452 |
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{(n+1/2)} |
| 453 |
\\ |
| 454 |
{\eta}^* & = & |
| 455 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
| 456 |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
| 457 |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
| 458 |
\end{eqnarray*} |
| 459 |
\\ |
| 460 |
In the hydrostatic case ($\epsilon_{nh}=0$), |
| 461 |
this allow to find ${\eta}^{n+1}$, according to: |
| 462 |
$$ |
| 463 |
\epsilon_{fs} {\eta}^{n+1} - |
| 464 |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{min}) |
| 465 |
{\bf \nabla}_h {\eta}^{n+1} |
| 466 |
= {\eta}^* |
| 467 |
$$ |
| 468 |
and then to compute (correction step): |
| 469 |
$$ |
| 470 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 471 |
- \beta \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 472 |
$$ |
| 473 |
|
| 474 |
The non-hydrostatic part is solved as described previously. |
| 475 |
\\ \\ |
| 476 |
N.B: |
| 477 |
\\ |
| 478 |
a) The non-hydrostatic part of the code has not yet been |
| 479 |
updated, %since it falls out of the purpose of this test, |
| 480 |
so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$. |
| 481 |
\\ |
| 482 |
b) To remind, the stability criteria with the Crank-Nickelson time stepping |
| 483 |
for the pure linear gravity wave problem in cartesian coordinate is: |
| 484 |
\\ |
| 485 |
$\star$~ $\beta + \gamma < 1$ : unstable |
| 486 |
\\ |
| 487 |
$\star$~ $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
| 488 |
\\ |
| 489 |
$\star$~ $\beta + \gamma \geq 1$ : stable if |
| 490 |
%, for all wave length $(k\Delta x,l\Delta y)$ |
| 491 |
$$ |
| 492 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
| 493 |
$$ |
| 494 |
$$ |
| 495 |
\mbox{with }~ |
| 496 |
%c^2 = 2 g H {\Delta t}^2 |
| 497 |
%(\frac{1-cos 2 \pi / k}{\Delta x^2} |
| 498 |
%+\frac{1-cos 2 \pi / l}{\Delta y^2}) |
| 499 |
%$$ |
| 500 |
%Practically, the most stringent condition is obtained with $k=l=2$ : |
| 501 |
%$$ |
| 502 |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
| 503 |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
| 504 |
$$ |
Parent Directory
| 
Revision Log
| 
Revision Graph