Selecting the time step

The time step for your simulation is selected as follows:

·         First you determine the grid spacing, Dx.

·         Secondly you decide on the maximum allowed Courant number, C_r, as described under Courant Number (p. 98).

·         Then you can determine the maximum time step, Dt_max, which can be used in the model from the definition of the Courant number:

(6.33)   

where c is the celerity (see Courant Number (p. 98) for a description). The time step to be used in the model, Dt, can then be chosen as a “con­venient” number not greater thanDt_max.

·         Finally you have to check that the Courant number based on the current speed, (the transport Courant number) C_r,U, instead of the wave celerity, is less than 1 for the time step chosen. If not, you must reduce the cho­sen time step. C_r,U is defined as

(6.34)   

where U,V is the current speed in the x-direction and y-direction. For all points within the model U and V must follow the above formula. As you have not yet carried out the simulation you will have to make an estimate and then check this after the simulation.

Please note that formally U and V in the above expression should be replaced by P/H and Q/H, respectively, where P and Q are the magnitudes of the fluxes in the x- and y- directions and H is the local water depth. Often the two expressions – based on current speeds or on the flux magnitudes – yield almost the same value, but care should be taken in models with large surface and/or bottom gradients.