A fast method for nonlinear three-dimensional free-surface waves

Fochesato, Christophe; Dias, Frédéric

doi:10.1098/rspa.2006.1706

Cited by 71 publications

(63 citation statements)

References 32 publications

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“…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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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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mentioning

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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“…As could be seen from the table, the QALE-FEM takes only 0.91h (or 54 minutes) to produce the results with acceptable errors in mass and energy (ε m = 0.1% and ε e = 0.26%, respectively). Even to achieve higher accuracy of ε m = 0.09% and ε e = 0.16% , which are smaller than those errors given by Fochesato & Dias [55], the CPU time taken by the QALE-FEM is only 1.8h (or 108 minutes). Therefore, for this particular case, the QALE-FEM can be at least 10 times faster than the fast BEM method.…”

Section: Computational Efficiencymentioning

confidence: 88%

“…The fast BEM method has successfully modelled 3D overturning solitary waves [55] and freak waves [56].…”

Section: Introductionmentioning

confidence: 99%

“…Although these applications have shown a good applicability of this model, its computational efficiency needs to be improved. For this purpose, Fochesato & Dias [55] introduced a fast multipole algorithm (FMA), referred as fast BEM method.…”

Section: Introductionmentioning

confidence: 99%

“…In the studies discussed above, the BVP is solved by using the BEM, either linear/low-order BEM (see, for instance, [45]), higher-order BEM (see, for example, [29]) or the fast BEM ( [55][56]). On the other hand, the FEM has been developed by Wu & Eatock Taylor [57] and Ma, Wu & Eatock Taylor [58][59] to solve fully nonlinear wave problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

QALE‐FEM for modelling 3D overturning waves

Yan

2009

Numerical Methods in Fluids

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This is the accepted version of the paper.This version of the publication may differ from the final published version. based on a fully nonlinear potential theory is presented in this paper. This development enables the QALE-FEM to deal with 3D (three dimensional) overturning waves over complex seabeds, which have not been considered since the method was devised by the authors of this paper in their previous works [1][2]. In order to tackle challenges associated with 3D overturning waves, two new numerical techniques are suggested. Permanent repository linkThey are the techniques for moving the mesh and for calculating the fluid velocity near overturning jets, respectively. The developed method is validated by comparing its numerical results with experimental data and results from other numerical methods available in the literature. Good agreement is achieved. The computational efficiency of this method is also investigated for this kind of wave, which shows that the QALE-FEM can be many times faster than other methods based on the same theory. Furthermore, 3Doverturning waves propagating over a non-symmetrical seabed or multiple reefs are simulated using the method. Some of these results have not been found elsewhere to the best of our knowledge.

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Phase‐Resolving Wave Modeling

Chalikov

2017

Encyclopedia of Maritime and Offshore Engineering

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Phase‐resolving modeling of sea waves is the mathematical modeling of surface waves including explicit reproduction of the sea surface and velocity field evolution. The general approach is based on equation of potential motion with a free boundary. Conformal transformation of coordinates is the most effective method of solution of two‐dimensional ( x–z ) periodic problem. For three dimensional models, several different approaches have been developed: boundary integral equation method (BIEM), method based on expansion of velocity potential in a vicinity of surface [high order spectral method (HOSM)], and method based on direct solution of three‐dimensional equation for velocity potential. These methods do not compete with each other because all of them have their advantages. The brief descriptions of each method are given as well as the examples of theirs applications. The investigations of nonperiodic problems are based on a direct solution of equation for velocity potential. The methods of including in the models of algorithms for parameterization of energy input and dissipation are discussed.

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A fast method for nonlinear three-dimensional free-surface waves

Cited by 71 publications

References 32 publications

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

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

QALE‐FEM for modelling 3D overturning waves

Phase‐Resolving Wave Modeling

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