A Whitham–Boussinesq long-wave model for variable topography

Vargas-Magaña, Rosa María; Panayotaros, Panayotis

doi:10.1016/j.wavemoti.2016.04.013

Cited by 14 publications

(16 citation statements)

References 31 publications

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“…Higher order expansions in β have been considered in the numerical studies of [15][16][17]. To avoid these longer expressions in simplified nonlocal shallow water equations, [32] proposed an ad-hoc approximation A G 0 of the linear Dirichlet-Neumann operator given by…”

Section: Water Wave Problem In Variable Depth and Approximate Dirichlmentioning

confidence: 99%

“…A consequence is that at high frequencies we recover the constant depth operator G(0, 0), under minimal assumptions on β. The matrix representation of the ad-hoc approximation A G 0 in the Fourier basis also becomes diagonal at high wave number, with diagonal entries approaching those of the infinite depth operator, see [32] for examples with smooth and piecewise constant depth profiles. However, the variable depth part seems to have weaker smoothing properties, e.g.…”

Section: Water Wave Problem In Variable Depth and Approximate Dirichlmentioning

confidence: 99%

“…We study some problems on normal modes in the linear water wave theory, in particular we compute transverse and longitudinal normal modes in infinite straight channels of constant bounded cross-section, considering special depth profiles for which there are known explicit or semi-explicit analytical solutions in the literature [13,14,21,23,28]. We compare the results to numerical normal modes obtained using simple approximations of the nonlocal variable depth Dirichlet-Neumann operator for the linear water wave equations, in particular a recently proposed ad-hoc approximation [32], as well as a first-order truncation of the systematic expansion in the depth proposed by Craig, Guyenne, Nicholls and Sulem [7]. Our main motivation is to test the simple approximations to the Dirichlet-Neumann operator as they are of interest in constructing simplified nonlocal shallow water models, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The study of linear normal modes is, therefore, a good test problem for comparing different approximations of the Dirichlet-Neumann operator for variable depth. In this paper we focus on simple approximations of the variable depth operator, such as the ad-hoc generalization of the constant depth operator for arbitrary depth proposed in [32]. This operator has some of the structural properties of the exact Dirichlet-Neumann operator, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

2019

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We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of 45 • and 60 • , see Lamb (Hydrodynamics. to numerical solutions obtained using approximations of the non-local Dirichlet-Neumann operator for linear waves, specifically an ad-hoc approximation proposed in Vargas-Magaña and Panayotaros (Wave Motion 65:156-174, 2016), and a first-order truncation of the systematic depth expansion by Craig et al. (Proc R Soc Lond A: Math, Phys Eng Sci 46: 2005). We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet-Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary. KeywordsLinear water waves · Whitham-Boussinesq model over variable topography · Dirichlet-Neumann operator · Normal modes on triangular straight channels · pseudodifferential operators A. A. Minzoni: Deceased. B R. M. Vargas-Magaña

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Section: Water Wave Problem In Variable Depth and Approximate Dirichlmentioning

confidence: 99%

Section: Water Wave Problem In Variable Depth and Approximate Dirichlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

2019

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“…In another recent work, Vargas-Magaña & Panayotaros [26] use the series expansion of Craig et al [9] and deduce a Whitham-Boussinesq model with topography. The numerical modeling is two dimensional.…”

Section: Introductionmentioning

confidence: 99%

A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

Andrade

Nachbin

2018

J. Fluid Mech.

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Surface water waves are considered propagating over highly variable non-smooth topographies. For this three dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modeling and evolution to the two dimensional free surface. The corresponding Fourier-type operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care and a Galerkin method is provided accordingly. One dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large amplitude, rapidly varying topography. An alternative, conformal mapping based, method is used for benchmarking. A two dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three dimensional DtN operator.

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A comparative study of bi-directional Whitham systems

Dinvay

Dutykh

Kalisch

2019

Applied Numerical Mathematics

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In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1,3,8,17,23,24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

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A Whitham–Boussinesq long-wave model for variable topography

Cited by 14 publications

References 31 publications

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

A comparative study of bi-directional Whitham systems

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