Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry?

Karczewska, Anna; Rozmej, P.

doi:10.1016/j.cnsns.2019.105073

Cited by 18 publications

(31 citation statements)

References 44 publications

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“…In the following, the authors apply the perturbative approach described in detail by Burde and Sergyeyev [2] and next extended in [3] to more complicated cases. In this method, one begins from zeroth-order solutions, then uses their properties in the calculation of corrections of the first order, and so on.…”

Section: Details Of Calculations In [1] Limited To First Ordermentioning

confidence: 99%

Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]

Rozmej

Karczewska

2021

Nonlinear Dyn

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The authors of the paper “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first-order (2+1)-dimensional equation. The equation has been obtained by applying the perturbation method Burde (J Phys A: Math Theor 46:075501, 2013) for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.

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Section: Details Of Calculations In [1] Limited To First Ordermentioning

confidence: 99%

Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]

Rozmej

Karczewska

2021

Nonlinear Dyn

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“…5 to determine the development of the wave height in the shoaling region. First formulate H = H (m) according to formula (24) given in Sect. 6.…”

Section: Shoaling Without Set-downmentioning

confidence: 99%

“…It is also possible to derive simpler KdV-type equations with variable bathymetry, such as shown in [17,20,40,58]. While some concerns with this approach were raised recently in [24], the authors of [40] found fair agreement of their hybrid spectral KdV-type model with shoaling experiments. A more recent approach is the inclusion of bathymetry in higher order, fully nonlinear or fully dispersive models, such as, for example, found in [13,14,59].…”

Section: Introductionmentioning

confidence: 99%

A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

Paulsen

Kalisch

2021

Water Waves

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In this article, we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg–de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-down $${\bar{\eta }}$$ η ¯ and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works.

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“…Generalizations of the KdV equation and of other dispersive one-wave models are discussed in some detail in [14,8,15]. For our purposes, these generalizations are useful as they allow to broaden the spectrum of benchmarks.…”

Section: Including the Bathymetric Effectsmentioning

confidence: 99%

Model order reduction strategies for weakly dispersive waves

Torlo¹,

Ricchiuto²

2021

Preprint

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We focus on the numerical modelling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM-KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction.

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Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry?

Cited by 18 publications

References 44 publications

Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]

Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]

A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

Model order reduction strategies for weakly dispersive waves

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