Dispersive Shallow Water Wave Modelling. Part IV: Numerical Simulation on a Globally Spherical Geometry

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

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

Cited by 10 publications

(21 citation statements)

References 48 publications

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“…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%

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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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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

Self Cite

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“…The reduction of spherical FNWD to WNWD models was illustrated, and several known models were recovered in this way. Finally, the fully nonlinear numerical results have been presented in Reference [18]. In particular, the importance of sphericity, Coriolis, and dispersion effects was thoroughly discussed.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the importance of sphericity, Coriolis, and dispersion effects was thoroughly discussed. The numerical algorithm employed in Reference [18] is based on the splitting approach: An elliptic equation is solved to determine the non-hydrostatic pressure component, and an evolution system of hyperbolic equations with source terms is solved to advance in time the velocity and water height variables. The algorithm was implemented as an explicit two-step predictor-corrector scheme.…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm was implemented as an explicit two-step predictor-corrector scheme. The volume of publication [18] did not allow the authors to describe all details of the employed numerical method. Henceforth, if we adopt dispersive models for tsunami simulations (It seems now that future generation tsunami modelling codes will include at least some dispersive effects), there is a clear need in robust numerical methods to solve hyperbolic balance equations.…”

Section: Introductionmentioning

confidence: 99%

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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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Discrete shallow water equations preserving symmetries and conservation laws

Дородницын

Капцов

2021

Journal of Mathematical Physics

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The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogs of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

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Dispersive Shallow Water Wave Modelling. Part IV: Numerical Simulation on a Globally Spherical Geometry

Cited by 10 publications

References 48 publications

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation 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

Discrete shallow water equations preserving symmetries and conservation laws

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