On the Efficient Numerical Simulation of Directionally Spread Surface Water Waves

Bateman, W. J. D.; Swan, Chris; Taylor, Paul H.

doi:10.1006/jcph.2001.6906

Cited by 108 publications

(94 citation statements)

References 24 publications

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“…These show excellent agreement with the analytical results above for a wide range of initial bandwidths, confirming at least that our analytical model works well for wave groups in the NLSE. & Taylor (2005) carried out fully nonlinear simulations of focusing wave groups in deep water, using the scheme developed by Bateman et al (2001). The simulations used a Gaussian wave packet as the initial conditions with a spectrum based on a JONSWAP spectrum with g = 3.3 and with a r.m.s.…”

Section: (A) Comparison With Nlsementioning

confidence: 99%

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water

2012

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The evolution of steep waves in the open ocean is nonlinear. In narrow-banded but directionally spread seas, this nonlinearity does not produce significant extra elevation but does lead to a large change in the shape of the wave group, causing, relative to linear evolution, contraction in the mean wave direction and lateral expansion. We use the nonlinear Schrödinger equation (NLSE) to derive an approximate analytical relationship for these changes in group shape. This shows excellent agreement with the numerical results both for the NLSE and for the full water wave equations. We also consider the application of scaling laws from the NLSE in terms of wave steepness and bandwidth to solutions of the full water wave equations. We investigate these numerically. While some aspects of water wave evolution do not scale, the major changes that a wave group undergoes as it evolves scale very well.

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Section: (A) Comparison With Nlsementioning

confidence: 99%

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water

2012

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“…This integral equation provides the summation of the Dirichlet-Neumann series operator discussed in Craig & Sulem (1993), Craig & Nicholls (2000) and Bateman et al (2001). Equations (4.6) and (4.2) (or (4.5)), are the closed system of water wave equations.…”

Section: The Non-dimensional Formmentioning

confidence: 99%

On a new non-local formulation of water waves

2006

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The classical equations of water waves are reformulated as a system of two equations, one of which is an explicit non-local equation, for the wave height and for the velocity potential evaluated on the free surface. Evaluation of the velocity potential as a function of the depth is not required in order to calculate the wave height and the velocity potential on the free surface. The non-local system yields integral relations related to mass and centre of mass, and is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney-Luke and nonlinear Schrödinger equations. Two-dimensional lumps with sufficient surface tension are obtained numerically. The extension of this non-local formulation to the case of a variable bottom is also presented.

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“…infinite depth [19,48,13,41,1], and related methods have been proposed to include the effects of an uneven bottom [33,46,4,28].…”

Section: Philippe Guyenne and David P Nichollsmentioning

confidence: 99%

“…The linear terms in (2.2) are solved exactly by an integrating factor technique [13,1]. The nonlinear terms are integrated in time using a fourth-order Runge-Kutta scheme with constant step size.…”

Section: Time Integrationmentioning

confidence: 99%

A High-Order Spectral Method for Nonlinear Water Waves over Moving Bottom Topography

Guyenne¹,

Nicholls²

2008

SIAM J. Sci. Comput.

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Abstract. We present a numerical method for simulations of nonlinear surface water waves over variable bathymetry. It is applicable to either two-or three-dimensional flows, as well as to either static or moving bottom topography. The method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving boundary quantities alone. A key component of this formulation is the Dirichlet-Neumann operator which, in light of its joint analyticity properties with respect to surface and bottom deformations, is computed using its Taylor series representation. We present new, stabilized forms for the Taylor terms, each of which is efficiently computed by a pseudospectral method using the fast Fourier transform. Physically relevant applications are displayed to illustrate the performance of the method; comparisons with analytical solutions and laboratory experiments are provided. 1. Introduction. Accurate modeling of surface water wave dynamics over bottom topography is of great importance to coastal engineers and has drawn considerable attention in recent years. Here are just a few applications: linear [16,33] and nonlinear wave shoaling [25,24,28,27], Bragg reflection [35,15,36,14,30,33,4], harmonic wave generation [3,17,18], and tsunami generation [37,22]. As the publications cited above attest, the character of coastal wave dynamics can be very complex: when entering shallow water, waves are strongly affected by the bottom through shoaling, refraction, diffraction, and reflection. Nonlinear effects related to wave-wave and wave-bottom interactions can cause wave scattering and depth-induced wave breaking. In turn, the resulting nonlinear waves can have a great influence on sediment transport and the formation of sandbars in nearshore regions. The presence of bottom topography also introduces additional space and time scales to the classical perturbation problem (see, e.g., [10]).Traditionally, the water wave problem on variable depth has been modeled using long-wave approximations such as the Boussinesq or shallow water equations [26,40]. See also [39] for a weakly nonlinear formulation for water waves over topography. In recent years, progress in both mathematical techniques and computer power has led to a rapid development of numerical models solving the full Euler equations. These can be divided into two main categories: boundary integral methods [2,24,49,7,20,27] and spectral methods. In particular, efficient spectral methods based on perturbation expansions have been developed for the computation of water waves on constant or

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On the Efficient Numerical Simulation of Directionally Spread Surface Water Waves

Cited by 108 publications

References 24 publications

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water

On a new non-local formulation of water waves

A High-Order Spectral Method for Nonlinear Water Waves over Moving Bottom Topography

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