Choice of numerical parameters in HOS-NWT

This section describes the main elements driving the choice of the numerical parameters in HOS-NWT. The model is based on pseudo-spectral formalism, which decompose the different surface spatial quantities on the normal modes of the domain.

Spatial/Modal discretization

Horizontal: n1 and n2

It is possible to provide general recommendations in the choice of the discretization in x-direction n1. However, as any numerical parameter, a convergence study should be carried out to test the influence of changing n1. One should take into account the following points:

• With a pseudo-spectral approach, the discretization defines the largest wavenumber that can be solved k_max =(n1-1)*π/xlen

• One wants to simulate a wavefield with a given peak wavenumber k_p (or eq. regular wavenumber k₀)

• In HOS-NWT one should use: k_max ≈ 20 k_p

For the discretization along y-direction n2, the choice should be done carefully after a convergence study. General recommendation cannot be provided since it is highly dependent on the directional spreading associated with the simulated wave field.

Vertical: n3

Similarly to the discretization along y-direction, it is complicated to provided general recommendation for the one along z-direction n3. A convergence study can be carried out since the results will be highly dependent on the wavemaker geometry: for instance with hinged wavemaker a more refined discretization is needed when d_hinge is a large part of the total water depth h. For ECN ocean engineering wave tank, results are usually converged with n3=17

Order of nonlinearity mHOS

The HOS method relies on an iterative procedure that is controlled by the so-called HOS order of nonlinearity mHOS. This parameter can be seen as a pure numerical parameter and consequently its effect can be characterized by a classical convergence study. However, regarding the HOS formulation, it is also possible to relate this numerical parameter to physical processes at play during wave propagation. Indeed, this order of nonlinearity corresponds to the nonlinear wave interactions between wave components that you take into account.

• mHOS=3 corresponds to third-order of nonlinearity, or equivalently the so-called four-wave interactions. With this set-up, the HOS model is equivalent to Zakharov equation for water waves. This is the minimum value suggested, if one focuses on nonlinear wave properties since it includes the most important nonlinear features at play during wave propagation

• mHOS=1 corresponds to a pure linear NWT (associated with i_wmk=1)

• For practical applications, this parameter is set in the range [3,8] and mHOS=5 is a standard value

Dealiasing p1 and p2

Pseudo-spectral methods face the possible issue of aliasing. A zero-padding procedure is set-up as dealiasing methodology in HOS-NWT, which is controlled thanks to the parameters p1 and p2. The following recommendations can be made

• Increased accuracy is achieved with total dealiasing p1=mHOS. This is at the expense of computational effort as well as numerical stability

• A typical optimal value for this parameter (robustness/accuracy) is p1=3

• For unidirectional configuration n2=1, one should use p2=1

• For multidirectional waves, one should use p2=p1

Temporal integration

The accuracy of the time integration scheme is controlled with the tolerance parameter toler. This is a numerical parameter, which convergence can be studied for a detailed study conducted with HOS-NWT. A value of toler=10^(-6) is suggested.