Dispersive Shallow Water Wave Modelling. Part III: Model Derivation on a Globally Spherical Geometry

Khakimzyanov, G. S.; Dutykh, Denys; Fedotova, Z. I.

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

Cited by 8 publications

(19 citation statements)

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“…Ideally, we would like to generalize the algorithm presented in this study for the SERRE-GREEN-NAGHDI (SGN) equations to the base model in its most general form (1.3), (1.4). In this way we would be able to incorporate several fully nonlinear shallow water models (discussed in Part I [91]) in the same numerical framework. It would allow the great flexibility in applications to choose and to assess the performance of various approximate models.…”

Section: Perspectivesmentioning

confidence: 99%

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

Khakimzyanov¹,

Dutykh²,

Gusev³

et al. 2018

CiCP

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In this paper we describe a numerical method to solve numerically the weakly dispersive fully nonlinear SERRE-GREEN-NAGHDI (SGN) celebrated model. Namely, our scheme is based on reliable finite volume methods, proven to be very efficient for the hyperbolic part of equations. The particularity of our study is that we develop an adaptive numerical model using moving grids. Moreover, we use a special form of the SGN equations where non-hydrostatic part of pressure is found by solving a linear elliptic equation. Moreover, this form of governing equations allows to determine the natural form of boundary conditions to obtain a well-posed (numerical) problem.

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Section: Perspectivesmentioning

confidence: 99%

“…In the following parts of this series of papers we shall discuss the derivation of the SGN equations on a sphere [91] and their numerical simulation using the finite volume method [93].…”

Section: Perspectivesmentioning

confidence: 99%

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

Khakimzyanov¹,

Dutykh²,

Gusev³

et al. 2018

CiCP

Self Cite

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“…The first steps in this direction have already been made in [34,35]. The derivation of shallow water equations on a sphere will be discussed in Part III [43].…”

Section: Perspectivesmentioning

confidence: 99%

“…the bottom is not necessarily flat. The (globally) spherical geometries will be discussed in some detail in Parts III & IV [43,44].…”

Section: Introductionmentioning

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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“…Such FNWD spherical equations (with the horizontal velocity defined on a certain surface inside the fluid bulk) have been derived in Reference [16]; however, only WNWD numerical results were presented. In a recent work [17], a systematic derivation of spherical rotating FNWD (and WNWD) models have been performed and the so-called base model was derived, which incorporates many other models as particular cases. The reduction of spherical FNWD to WNWD models was illustrated, and several known models were recovered in this way.…”

Section: Introductionmentioning

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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Dispersive Shallow Water Wave Modelling. Part III: Model Derivation on a Globally Spherical Geometry

Cited by 8 publications

References 0 publications

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

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

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

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

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