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Vertical pressure gradient

Space discretization of the governing equations is carried out in a finite volume/finite difference fashion. A staggered grid arrangement is used in which the velocity components are located at the center of the cell faces (see Figure 5.2). The water level is located at cell center. Concerning the non-hydrostatic pressure, two layouts to assign this unknown to grid points are employed. This variable can be given either at the cell center or at the layer interface. The former is called the standard layout, while the latter one is called the box layout; see Figure 5.2.

Figure 5.2: Applied arrangements of the unknowns in a staggered grid: (a) standard layout and (b) box layout. Meaning of the unknowns: u is the horizontal velocity, w is the vertical velocity and q is the non-hydrostatic pressure.

The choice depends on the discretization of the vertical pressure gradient, namely, explicit central differences referring as the classical case and the implicit Keller-box or compact scheme, respectively. This compact scheme allows straightforward implementation of the zero pressure boundary condition at the free surface without the need for special attention at interior points near that surface. Moreover, the discretization error is four to six times smaller than the error of classical central differences of the same order and involving the same number of vertical grid points. Hence, use of the compact scheme allows a very few number of vertical grid points with relative low numerical dispersion and dissipation, thereby enhancing the accuracy of the frequency dispersion for relative short waves up to an acceptable level, see Table 5.1.

At very low vertical resolution (one or two layers), the Keller-box scheme gives good dispersive properties. At high vertical resolutions, however, the standard layout is preferable because it appears to be more robust while its dispersion characteristics are then usually sufficiently accurate.

To summarize, for wave simulations with 5 layers or less, the Keller-box scheme using the box layout is recommended, while for simulations with typically 10-20 layers, the classical central differencing employing the standard layout is preferred. See command NONHYDrostatic STANdard|BOX.

Related to this choice, it might be useful to specify the preconditioner for solving the Poisson pressure equation. Two options are available: ILU and ILUD. For a robust solution, the ILU preconditioner is preferred. This choice might be a good one for applications where high and short waves are involved, or irregular beds with steep slopes (e.g. weir, breakwater, quay, jetty), or when relatively large number of layers (> 30) are involved. On the other hand, the ILUD preconditioner is a better choice to get an efficient solution (e.g. parallel computing). See command NONHYDrostatic ... PREConditioner ILUD|ILU.

next up previous index
Next: Momentum conservation Up: Numerical parameters Previous: Time step   Index
The SWASH team 2017-04-06