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##

Moving shorelines

For the calculation of wave runup and rundown on the beach, use of a moving boundary condition is required. The method used in SWASH
to track the moving shoreline amounts to ensure non-negative water depths. For a one-dimensional case, one can show that if

1

and if a first order upwind scheme is applied to the global continuity equation, we shall have non-negative
water depths at every time step; see Stelling and Duinmeijer (2003) for a proof.
Hence, flooding never happens faster than one grid size per time step, which is physically correct.
This implies that the calculation of the dry areas does not need any special feature.
For this reason, no complicated drying and flooding procedures are required.
Additionally, the shoreline motion in the swash zone can be simulated in a natural manner.

For computational efficiency, the model equations are not solved and the velocities are set to zero when the water
depth is below a threshold value (see command `SET DEPMIN`). Its default value is 0.05 mm.
However, a higher threshold value may be chosen for scaling reasons. For instance, at the scale of a field site,
a value of 1 mm is an appropriate choice. (As a matter of fact, the value of 0.05 mm is a suitable one under laboratory conditions.)
This will also relax the time step to some extent in case of explicit time stepping.
For a large-scale ocean simulation, a threshold value of 1 cm is probably more effective than 0.05 mm.
Be careful when choosing a too high value as this may negatively influence mass conservation.

To achieve second order accuracy, the so-called MUSCL limiter may be employed (see command `DISCRET CORRDEP`).

Since the CFL condition, Eq. (5.1), holds this implies that ensuring non-negative water depths does not lead to a new time step restriction.

For some two-dimensional cases, however,
ensuring non-negative water depths might lead to a time step restriction which appears to be more restrictive than the usual CFL condition.
An example is the case where locally all velocities are directed outward of a grid cell.
Nevertheless, such a case is rarely encountered and usually the time step is restricted by the Courant number based on the stability criterion.

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The SWASH team 2017-04-06