On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves

Abstract

This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet–Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M , and spatial resolution, N , and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

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

“…Our experiments with different parameter choices show that for kh = 0.3 we could get a match similar to the one in Figure 8(A) for wave heights up to about H / H max = 0.8 and to obtain this, we had to use less damping () and a significantly smaller time step () than used for . In comparison, we have recently shown (see Klahn et al 11 ) that for kh = 0.3 the HOS method of Dommermuth and Yue 9 only allows stable time integration of waves with if a comparable accuracy is to be achieved. When seen in this light, the time integration of the present method may therefore be considered to be quite robust indeed.…”

“…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. In fact, Ducrozet et al 12 have shown that their implementation of the HOS method has an execution time which is typically two orders of magnitude smaller than that of the finite difference method of Ensig‐Karup et al 4 when dealing with long‐time propagation of regular wave trains and requiring the final phase error to be less than 0.1 degrees (see figure 13 of Ducrozet et al).…”

Direct Simulation of the Surface Manifestation of Internal Gravity Waves with a Wave–Current Interaction Model

“…A wave-current interaction model similar to ours has been presented by a number of studies, although the surface current is assumed to be stationary (Wu 2004;Wang et al 2018;Pan 2020;Ducrozet et al 2021). The present work considers a propagating surface current such that the corresponding correction is made in the wave-current interaction terms, and an efficient highorder spectral (HOS) method is used to solve the model equations (Klahn et al 2020). In addition to the distinct features of the numerical model, this study is unique because, as far as we are aware, it is the first direct simulation of the interaction of a random surface wave field with internal waves in which the results are compared to observations in a real, field-scale setting.…”

On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves

“…The high-order spectral (HOS) method (Dommermuth & Yue 1987;West et al 1987), Boussinesq-type formulations (Wei et al 1995;Agnon, Madsen & Schäffer 1999), volumetric methods (Engsig-Karup, Bingham & Lindberg 2009;Bihs et al 2020) and the fast computational method developed by Clamond & Grue (2001) are a few examples of these that can account for waves up to arbitrary order in wave steepness. In terms of the numerical efficiency for a given level of accuracy, it is without doubt that the HOS method is the preferred choice compared with the aforementioned alternatives (Klahn, Madsen & Fuhrman 2020). Two aspects have especially contributed to its high efficiency: (i) it permits an explicit method for the time integration and vertical velocity on the free water surface, and (ii) it takes advantage of spectral methods for numerical computations (Dommermuth & Yue 1987;Ducrozet et al 2016).…”