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Next: 2.9.2 Non-linear free-surface
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2.9.1 Crank-Nickelson barotropic time stepping
The full implicit time stepping described previously is
unconditionally stable but damps the fast gravity waves, resulting in
a loss of potential energy. The modification presented now allows one
to combine an implicit part (
) and an explicit part
(
) for the surface pressure gradient () and
for the barotropic flow divergence ().
For instance,
is the previous fully implicit scheme;
is the non damping (energy conserving), unconditionally
stable, Crank-Nickelson scheme;
or
corresponds to the forward - backward scheme that conserves energy but is
only stable for small time steps.
In the code,
are defined as parameters, respectively
implicSurfPress, implicDiv2DFlow. They are read from
the main parameter file "data" and are set by default to 1,1.
Equations 2.17 -
2.22 are modified as follows:
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(2.76) |
where:
In the hydrostatic case (
), allowing us to find
, thus:
and then to compute (CORRECTION_STEP):
Notes:
- The RHS term of equation 2.78
corresponds the contribution of fresh water flux (P-E)
to the free-surface variations (
,
useRealFreshWater=TRUE in parameter file data).
In order to remain consistent with the tracer equation, specially in
the non-linear free-surface formulation, this term is also
affected by the Crank-Nickelson time stepping. The RHS reads:
- The non-hydrostatic part of the code has not yet been
updated, and therefore cannot be used with
.
- The stability criteria with Crank-Nickelson time stepping
for the pure linear gravity wave problem in cartesian coordinates is:
-
: unstable
-
and
: stable
-
: stable if
with
Next: 2.9.2 Non-linear free-surface
Up: 2.9 Variants on the
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