An operator expansion formalism for nonlinear surface waves over variable
 depth

Smith, R. C.

doi:10.1017/s0022112098001219

Cited by 42 publications

(39 citation statements)

References 21 publications

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“…(9). An alternative and elegant scheme for the calculation of the operators G n ( f ) has been used by a number of authors in various applications, including the study of gravity water waves (9,15,18,27,30) and ocean scattering (17,(19)(20)(21)(22)(23)(24)29). The method works directly with the DNO without reference to the bulk potential and has thus been termed the Operator Expansion (OE) Method.…”

Section: The Field Expansion and Operator Expansion Methodsmentioning

confidence: 99%

“…These algorithms are based on the derivation of (low-or high-order) series representing the DNO in powers of a parameter measuring deviations from a separable, easily solvable geometry (e.g., planar, spherical, ellipsoidal) for which the DNO can be found explicitly. As has been demonstrated (9,11,21,27), perturbation methods can lead very efficiently to accurate results within their domain of applicability. More importantly perhaps, and in contrast with alternative methods (e.g., finite elements or surface potentials), the implementation and performance of perturbative approaches do not depend strongly on the spatial dimension, a feature that makes them particularly attractive for three-dimensional calculations.…”

Section: Introductionmentioning

confidence: 98%

“…Specifically, we considered two main implementations of these methods which include the well-known Operator Expansion (OE) Method used by Milder et al (17,(19)(20)(21)(22)(23)(24)29) in the context of the boundary value problems of electromagnetic and acoustic scattering, and by a number of authors (9,15,18,27,30) in the study of the classical free boundary model of gravity water waves (see Section 2). We also considered a different scheme, which we will refer to as the Field Expansion (FE) Method, that was proposed by Dommermuth and Yue (11), again in the context of water wave simulations, and that is based on the expansion of the full field quantities in bulk (see Section 2).…”

Section: Introductionmentioning

confidence: 99%

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Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Nicholls¹,

Reitich²

2001

Journal of Computational Physics

117

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In this paper we present results on the stability of perturbation methods for the evaluation of Dirichlet-Neumann operators (DNO) defined on domains that are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition, have long been recognized to constitute efficient and accurate means for the approximation of DNO and, in fact, several numerical implementations have been previously proposed. Inspired by our recent analytical work, here we demonstrate that the convergence of these algorithms is, quite generally, limited by numerical instability. Indeed, we show that these standard perturbative methods for the calculation of DNO suffer from significant ill-conditioning which is manifest even for very smooth boundaries, and whose severity increases with boundary roughness. Moreover, and again motivated by our previous work, we introduce an alternative perturbative approach that we show to be numerically stable. This approach can be interpreted as a reformulation of classical perturbative algorithms (in suitable independent variables), and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series. It can also be related to classical (preconditioned) spectral approaches and, as such, it retains, in finite arithmetic, the spectral convergence properties of classical perturbative methods, albeit at a higher computational cost (as it does not take advantage of possible dimensional reductions). Still, as we show, an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high-order harmonics and /or perturbation coefficients of the solution.

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Section: The Field Expansion and Operator Expansion Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Nicholls¹,

Reitich²

2001

Journal of Computational Physics

117

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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%

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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“…Guyenne & Nicholls [36] studied two-dimensional gravity waves over plane slopes (bathymetry) which required a generalization of the DNO to include bottom topography. Please see Smith [78]; Craig, Guyenne, Nicholls, and Sulem [22]; and Nicholls and Taber [63] for more details on this extension. Recently, Craig, Guyenne, Hammack, Henderson, and Sulem [20] used this approach to revisit the problem of solitary wave interactions and compared the results with those of wave-tank experiments.…”

Section: Boundary Perturbation Methods For the Initial Value Problemmentioning

confidence: 99%

Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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Key words Water waves, free-surface fluid flows, ideal fluid flows, boundary perturbation methods, spectral methods.The most successful equations for the modeling of ocean wave phenomena are the freesurface Euler equations. Their solutions accurately approximate a wide range of physical problems from open-ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem." Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed.

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An operator expansion formalism for nonlinear surface waves over variable depth

Cited by 42 publications

References 21 publications

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

Stability of High-Order Perturbative Methods for the Computation of Dirichlet–Neumann Operators

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

Boundary Perturbation Methods for Water Waves

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