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Time step

The time integration is of explicit type and thus requires strict confirmity of stability criteria for a stable solution. The well-known CFL condition for 1D problems is given by

Cr = $\displaystyle {\frac{{\Delta t \left(\sqrt{gd} + \vert u\vert\right)}}{{\Delta x}}}$ $\displaystyle \leq$ 1 (5.1)
with $ \Delta$x the mesh width, $ \Delta$t the time step, u the flow velocity, and Cr the Courant number. For a 2D problem, however, the following CFL condition is employed

Cr = $\displaystyle \Delta$t $\displaystyle \left(\vphantom{ \sqrt{g d} + \sqrt{u^2 + v^2} }\right.$$\displaystyle \sqrt{{g d}}$ + $\displaystyle \sqrt{{u^2 + v^2}}$$\displaystyle \left.\vphantom{ \sqrt{g d} + \sqrt{u^2 + v^2} }\right)$ $\displaystyle \sqrt{{\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2}}}$ $\displaystyle \leq$ 1

A dynamically adjusted time step controlled by the Courant number in a user prescribed range is implemented in SWASH as follows. The actual maximum of the Courant number over all wet grid points is determined. The time step is halved when this number becomes larger than a preset constant Crmax < 1, and the time step is doubled when this number is smaller than another constant Crmin, which is small enough to be sure the time step can be doubled. Usually, Crmin is set to 0.2, while the maximum Courant number Crmax is specified in the range of 0.5 to 0.8. It is advised not to choose a value higher than 0.8 since nonlinear processes, e.g. wave breaking and wave-wave interactions, can affect the stability condition. For high, nonlinear waves, or wave interaction with structures with steep slopes (e.g. jetties, quays), a Courant number of 0.5 is advised.


next up previous index
Next: Vertical pressure gradient Up: Numerical parameters Previous: Duration of simulation   Index
The SWASH team 2017-04-06