Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

Khakimzyanov, G. S.; Dutykh, Denys; Gusev, Oleg; Shokina, Nina

doi:10.4208/cicp.oa-2016-0179b

Cited by 9 publications

(39 citation statements)

References 138 publications

(169 reference statements)

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“…Finally, the models presented in this study do not account for the dispersion of surface waves during their long-time propagation. However, using ideas developed in [14,15], it is possible to extend the proposed model to take into account these dispersion features in shallow water waves induced by elastic deformations in the bed topography.…”

Section: Discussionmentioning

confidence: 99%

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

Al-Ghosoun¹

2021

CiCP

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We propose a coupled model to simulate shallow water waves induced by elastic deformations in the bed topography. The governing equations consist of the depth-averaged shallow water equations including friction terms for the water freesurface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and friction forces are considered for the elastostatic equations whereas bathymetric forces are accounted for in the shallow water equations. The focus in the present study is on the development of a simple and accurate representation of the interaction between water waves and bed deformations in order to simulate practical shallow water flows without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and free-surface waves are investigated in details. Numerical simulations are carried out for several test examples on shallow water waves induced by sudden changes in the bed. The proposed computational model has been found to be feasible and satisfactory.

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Section: Discussionmentioning

confidence: 99%

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

Al-Ghosoun¹

2021

CiCP

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“…The "Holy Grail" in choosing the monitor function is to achieve ideally the reduction of the error in orders of magnitude when the discretization step ∆x → 0 , as illustrated in Reference [45]. The main advantages of the proposed approach include The present manuscript sheds also some light on the hyperobolic part of the splitting approach we used earlier to simulate the fully nonlinear and weakly dispersive wave propagations [46].…”

Section: Introductionmentioning

confidence: 90%

“…Finally, there is another goal behind this study. Ultimately, we would like to generalize moving grid methods to some dispersive wave equations [92], which are going to be addressed with the operator splitting approach along the lines sketched in Reference [46]. In this way, the hyperbolic part would be addressed with the methods outlined above.…”

Section: Perspectivesmentioning

confidence: 99%

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

et al. 2019

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In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

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“…We underline the fact that the last formula is accurate to the order O(µ 4 ). This formula will be used in [45] in order to reconstruct the pressure field under a solitary wave, which undergoes some nonlinear transformations. In order to obtain an evolution equation for the approximate horizontal velocityū(x, t) we integrate over the vertical coordinate equation (2.8):…”

Section: Long Wave Approximationmentioning

confidence: 99%

“…The numerical discretization of the derived above equations on moving adaptive grids will be considered in details in the companion paper [45] (Part II), while the numerical simulation of shallow water waves on a sphere will be considered in Part IV of this series of papers [44].…”

Section: Perspectivesmentioning

confidence: 99%

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

Khakimzyanov¹,

Dutykh²,

Fedotova³

et al. 2018

CiCP

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Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space arXiv.org / hal Abstract. In this paper we review the history and current state-of-the-art in modelling of long nonlinear dispersive waves. For the sake of conciseness of this review we omit the unidirectional models and focus especially on some classical and improved Boussinesqtype and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

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Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

Cited by 9 publications

References 138 publications

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

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