On the influence of wave reflection on shoaling and breaking solitary Waves

Senthilkumar, Amutha

doi:10.3176/proc.2016.4.06

“…Similar Boussinesq-type systems have been derived in [34]. It is noted that (6) cannot be recovered from the analogous regularized Boussinesq systems derived in [34], even in one-dimensional case by choosing appropriate coefficients, [42].…”

Section: Mathematical Modelsmentioning

confidence: 94%

“…between the unknowns η and w and the partial differential equation Solving the equation ( 40) for w we recover the free-surface elevation from (39). In order to solve (40) we employ again the standard Galerkin finite element method: seek w h ∈ U r 2 h satisfying the equation written in the weak form (42) L h (w h , χ) = (N (w h ), χ), for all χ ∈ U r 2 h , where L h (w, χ) is the bilinear form…”

Section: Appendix a The Petviashvili Iterationmentioning

confidence: 99%

“…Non-slip wall boundary conditions can be restrictive while the computation of the tangential component of the velocity on the boundary of the domain can be rather complicated. On the other hand, such BBM-BBM type system was shown to have very good performance in studies of water waves over variable bottom [42,28,34] and for this reason certain improvements can be made so as to make slip-wall boundary conditions easily applicable.…”

Section: Introductionmentioning

confidence: 99%

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Boussinesq-Peregrine water wave models and their numerical approximation

Katsaounis

¹

,

Mitsotakis

²

,

Sadaka

³

2020

Journal of Computational Physics

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In this paper we consider the numerical solution of Boussinesq-Peregrine type systems by the application of the Galerkin finite element method. The structure of the Boussinesq systems is explained and certain alternative nonlinear and dispersive terms are compared. A detailed study of the convergence properties of the standard Galerkin method, using various finite element spaces on unstructured triangular grids, is presented. Along with the study of the Peregrine system, a new Boussinesq system of BBM-BBM type is derived. The new system has the same structure in its momentum equation but differs slightly in the mass conservation equation compared to the Peregrine system. Further, the finite element method applied to the new system has better convergence properties, when used for its numerical approximation. Due to the lack of analytical formulas for solitary wave solutions for the systems under consideration, a Galerkin finite element method combined with the Petviashvili iteration is proposed for the numerical generation of accurate approximations of line solitary waves. Various numerical experiments related to the propagation of solitary and periodic waves over variable bottom topography and their interaction with the boundaries of the domains are presented. We conclude that both systems have similar accuracy when approximate long waves of small amplitude while the Galerkin finite element method is more efficient when applied to BBM-BBM type systems.

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“…Similar Boussinesq-type systems have been derived in [30]. It is noted that (12) cannot be recovered from the analogous regularized Boussinesq systems derived in [30], even in one-dimensional case [39] by choosing appropriate coefficients. Assuming that D is very smooth, i.e.…”

Section: 3mentioning

confidence: 95%

“…Non-slip wall boundary conditions can be restrictive and on the other hand the computation of the tangential component of the velocity on the boundary of the domain can be rather complicated. On the other hand, such BBM-BBM type system was shown to have very good performance in studies of water waves over variable bottom [39,24,30] and for this reason certain improvements can be made so as to make slip-wall boundary conditions easily applicable.…”

Section: Introductionmentioning

confidence: 99%

Boussinesq-Peregrine water wave models and their numerical approximation

Katsaounis,

Mitsotakis,

Sadaka

2020

Preprint

0

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In this paper we consider the numerical solution of Boussinesq-Peregrine type systems by the application of the Galerkin finite element method. The structure of the Boussinesq systems is explained and certain alternative nonlinear and dispersive terms are compared. A detailed study of the convergence properties of the standard Galerkin method, using various finite element spaces on unstructured triangular grids, is presented. Along with the study of the Peregrine system, a new Boussinesq system of BBM-BBM type is derived. The new system has the same structure in its momentum equation but differs slightly in the mass conservation equation compared to the Peregrine system. Further, the finite element method applied to the new system has better convergence properties, when used for its numerical approximation. Due to the lack of analytical formulas for solitary wave solutions for the systems under consideration, a Galerkin finite element method combined with the Petviashvili iteration is proposed for the numerical generation of accurate approximations of line solitary waves. Various numerical experiments related to the propagation of solitary and periodic waves over variable bottom topography and their interaction with the boundaries of the domains are presented. We conclude that both systems have similar accuracy when approximate long waves of small amplitude while the Galerkin finite element method is more effective when applied to BBM-BBM type systems.

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A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

Israwi

¹

,

Kalisch

²

,

Katsaounis

³

et al. 2021

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The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

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On the influence of wave reflection on shoaling and breaking solitary Waves

Cited by 4 publications

References 36 publications

Boussinesq-Peregrine water wave models and their numerical approximation

Boussinesq-Peregrine water wave models and their numerical approximation

Boussinesq-Peregrine water wave models and their numerical approximation

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

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