A new σ‐transform based Fourier‐Legendre‐Galerkin model for nonlinear water waves

Abstract

This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the-transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth-order Runge-Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.

Simulation of three‐dimensional nonlinear water waves using a pseudospectral volumetric method with an artificial boundary condition

“…Our results for u ( z ) below the wave crest are shown in Figure 6 together with the best results we could obtain using the stream function method of Rienecker and Fenton. The figure shows a close match between the two methods for all values of kh and H / H max , and we note that besides our own recent work (see Klahn et al 8 ), seemingly nobody has attempted or been able to compute the profiles for the waves with H / H max = 0.99. The relative difference between the results for u / c at the crest of our method and the stream function method is shown in Table 3 and from these we conclude that the difference between our method and the stream function method is smaller than the difference between the method of Engsig‐Karup et al and the stream function method by about a factor 1000 when kh = 1 and about a factor 625 when kh = 10.…”

“…The opposite is found to be the case. Second, when we compare the stability performance of the present model to the performance of our recent Galerkin model (see Klahn et al 8 ), which in essence is a two‐dimensional version of the present model that treats all nonlinear products in a completely aliasing‐free fashion, we see almost no difference in stability properties. At this stage, we do not know what causes the instability for increasing nonlinearity and resolution.…”

“…The method was first tested on steady nonlinear wave trains and we have shown that the method is capable of computing the vertical surface velocity with a relative error as low as 10 −12 for practically all water depth and relative wave heights. In addition, it was shown that if the vertical velocity was to be computed with an error less than some specified tolerance, only about half the degrees of freedom were needed in this method compared to the Galerkin method of Klahn et al 8 when . As such, using the artificial boundary condition does in fact enable substantial computational savings when the water depth is comparable in magnitude to the wavelength.…”

“…Besides comparing the present method to itself, it is of course also interesting to compare it to other methods in order to see the resolution needed to achieve w s with a certain accuracy. For that reason, we have computed and using our recent Galerkin method (see Klahn et al 8 ) which discretizes the entire fluid domain instead of using an artificial boundary condition as done here but otherwise may be though of as an alias‐free two‐dimensional version of the present method. We have found that, despite the fact that the Galerkin method is free of aliasing errors, the required horizontal resolution is almost exactly the same as for the present method for all water depths.…”

“…The research into these methods presumably began with the work of Li and Fleming, 2 who developed a finite difference‐based multigrid method for three‐dimensional wave fields, and since then, many different authors have used a volumetric approach for wave simulations. For example, Bingham and Zhang, 3 Engsig‐Karup et al, 4 Christiansen et al, 5 Yates and Benoit, 6 Raoult et al, 7 and Klahn et al 8 have all employed volumetric methods, and as a consequence it is now well established that this approach can handle variable bathymetry without approximations and that it leads to a stable computation of surface velocities as well as velocity profiles. These attractive features come at the prize of a substantially more involved computational procedure than, for example, that of the classic high‐order spectral method derived independently by Dommermuth and Yue 9 and West et al 10 Although the HOS method is not capable of dealing with very steep waves (see, e.g., Klahn et al 11 for an investigation of its range of applicability), it is as a starting point much more efficient than the volumetric methods when the degree of nonlinearity is not too large.…”