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Multilayer Shallow Water Research Articles

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Overview
49 Articles

Published in last 50 years

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Articles published on Multilayer Shallow Water

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A multilayer shallow water model for tsunamis and coastal forest interaction

Models and numerical methods of the impact of tsunamis on coastal forests are of vital importance for exploring the potential of coastal vegetation as a means of mitigation. Such a model is formulated as a multilayer shallow water system based on a free-surface formulation of the Euler equations for an ideal fluid. Specifically, the Euler equations are approximated by a layer averaged non-hydrostatic (LDNH) approach involving linear pressures and piecewise constant velocities. Furthermore, based on [Iimura and Tanaka, Ocean Eng. 54 (2012) 223–232] drag forces, inertia forces, and porosity are added to model the interaction with the forest. These ingredients are specified in a layer-wise manner. Thus, the vertical features of the forest are described with higher accuracy than within a single-layer approach. Projection methods for the non-hydrostatic pressure in conjunction with polynomial viscosity matrix finite volume methods [Castro and Fernández-Nieto, SIAM J. Sci. Comput. 34 (2012) A2173–A2196] are employed for the numerical solution of the multilayer model, that is for the propagation of tsunamis and coastal flooding. Experimental observations and field data are used to validate the model. In general good agreement is obtained.

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  • Journal IconESAIM: Mathematical Modelling and Numerical Analysis
  • Publication Date IconMar 1, 2025
  • Author Icon Raimund Bürger + 2
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A novel framework of the lattice Boltzmann model for multilayer shallow water systems

This study proposes a novel framework of the lattice Boltzmann model for multilayer shallow water equations, considering the mass and momentum exchanges between layers (LABMSWE+). Compared with the original LABMSWE model consisting of N two-dimensional lattice Boltzmann method for shallow water equation (LABSWE) models, the new model includes 1+N LABSWE models. The singular LABSWE model with unit relaxation time is introduced to update the total water depth, and thus, the layer water depths can be obtained explicitly through the fixed layer ratios. The N-layer LABSWE models with the multiple-relaxation-time operator evolve the layer velocities. These two modules are coupled by the total water depth and depth-averaged velocities. The constructed model avoids the freely variable layer thicknesses, which is considered as the main source of the instability. In addition, the mass exchanges enable this model to simulate vertical circulation flows, which are beyond the application of the LABMSWE model. Several numerical tests are then conducted to validate the proposed model. The results show that it exactly satisfies the C-property. In addition, the central difference scheme is more stable and accurate than the upwind and nonequilibrium schemes in the computing of the mass exchanges. The numerical results have an excellent agreement with analytical solutions and reference data, while some unstable and nonphysical results are obtained by the original LABMSWE model. Moreover, the computational time is about 40%–60% of that for the MIKE3, a finite volume solver for the three-dimensional shallow water equations by the Danish Hydraulic Institute.

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  • Journal IconPhysics of Fluids
  • Publication Date IconSep 1, 2024
  • Author Icon Zhiming Ru + 3
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Approximating a continuously stratified hydrostatic system by the multi-layer shallow water system

In this article we consider the multi-layer shallow water system for the propagation of gravity waves in density-stratified flows, with additional terms introduced by the oceanographers Gent and McWilliams ( Journal of Physical Oceanography 20 ( 1990 ) 150–155) in order to take into account large-scale isopycnal diffusivity induced by small-scale unresolved eddies. We establish a bridge between the multi-layer shallow water system and the corresponding system for continuously stratified flows, that is the incompressible Euler equations with eddy-induced diffusivity under the hydrostatic approximation. Specifically we prove that, under an assumption of stable stratification, sufficiently regular solutions to the incompressible Euler equations can be approximated by solutions to multi-layer shallow water systems as the number of layers, N , increases. Moreover, we provide a convergence rate of order 1 / N 2 . A key ingredient in the proof is a stability estimate for the multi-layer system which relies on suitable energy estimates mimicking the ones recently established by Bianchini and Duchêne (Bianchini and Duchêne ( 2024 )) on the continuously stratified system. This requires to compile a dictionary that translates continuous operations (differentiation, integration, etc. ) into corresponding discrete operations.

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  • Journal IconAsymptotic Analysis
  • Publication Date IconAug 7, 2024
  • Author Icon Mahieddine Adim
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High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

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  • Journal IconJournal of Computational Physics
  • Publication Date IconJul 29, 2024
  • Author Icon Zhihao Zhang + 2
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A multilayer shallow water model for polydisperse reactive sedimentation

A multilayer shallow water model for polydisperse reactive sedimentation

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  • Journal IconApplied Mathematical Modelling
  • Publication Date IconJun 19, 2024
  • Author Icon Julio Careaga + 1
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A fully coupled dynamic water-mooring line system: Numerical implementation and applications

A fully coupled dynamic water-mooring line system: Numerical implementation and applications

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  • Journal IconOcean Engineering
  • Publication Date IconJan 20, 2024
  • Author Icon Xiangcou Zheng + 2
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A semi-Lagrangian Splitting framework for the simulation of non-hydrostatic free-surface flows

A semi-Lagrangian Splitting framework for the simulation of non-hydrostatic free-surface flows

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  • Journal IconOcean Modelling
  • Publication Date IconNov 30, 2023
  • Author Icon Andreas Alexandris-Galanopoulos + 2
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A flexible z-layers approach for the accurate representation of free surface flows in a coastal ocean model (SHYFEM v. 7_5_71)

Abstract. We propose a discrete multilayer shallow water model based on z-layers, which, thanks to the insertion and removal of surface layers, can deal with an arbitrarily large tidal oscillation independently of the vertical resolution. The algorithm is based on a classical two-step procedure used in numerical simulations with moving boundaries (grid movement followed by a grid topology change, that is, the insertion/removal of surface layers), which avoids the appearance of surface layers with very small or negative thickness. With ad hoc treatment of advection terms at nonconformal edges that may appear owing to insertion/removal operations, mass conservation and the compatibility of the tracer equation with the continuity equation are preserved at a discrete level. This algorithm called z-surface-adaptive, can be reduced, as a particular case when all layers are moving, to the z-star coordinate. With idealized and realistic numerical experiments, we compare the z-surface-adaptive against z-star and we show that it can be used to simulate coastal flows effectively.

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  • Journal IconGeoscientific Model Development
  • Publication Date IconNov 28, 2023
  • Author Icon Luca Arpaia + 3
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A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-LobattoLegendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

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  • Journal IconAdvances in Applied Mathematics and Mechanics
  • Publication Date IconJun 1, 2022
  • Author Icon Nouh Izem Nouh Izem + 1
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Internal solitary waves with trapped cores in multilayer shallow water

Internal solitary waves with trapped cores in multilayer shallow water

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  • Journal IconTheoretical and Mathematical Physics
  • Publication Date IconMay 1, 2022
  • Author Icon V Yu Liapidevskii + 1
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An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

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  • Journal IconJournal of Scientific Computing
  • Publication Date IconDec 17, 2021
  • Author Icon E Guerrero Fernández + 3
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Parallel exponential time differencing methods for geophysical flow simulations

Parallel exponential time differencing methods for geophysical flow simulations

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  • Journal IconComputer Methods in Applied Mechanics and Engineering
  • Publication Date IconSep 20, 2021
  • Author Icon Rihui Lan + 4
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A surrogate model for efficient quantification of uncertainties in multilayer shallow water flows

A surrogate model for efficient quantification of uncertainties in multilayer shallow water flows

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  • Journal IconEnvironmental Modelling & Software
  • Publication Date IconAug 24, 2021
  • Author Icon Alia Al-Ghosoun + 2
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Flexible and efficient discretizations of multilayer models with variable density

Flexible and efficient discretizations of multilayer models with variable density

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  • Journal IconApplied Mathematics and Computation
  • Publication Date IconMar 6, 2021
  • Author Icon José Garres-Díaz + 1
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Quasi-Linear Equations of Dynamics of Internal Solitary Waves in Multilayer Shallow Water

Quasi-Linear Equations of Dynamics of Internal Solitary Waves in Multilayer Shallow Water

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  • Journal IconПрикладная механика и техническая физика
  • Publication Date IconJan 1, 2021
  • Author Icon V Yu Liapidevskii + 2
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A Multilayer Shallow Water Approach for Polydisperse Sedimentation with Sediment Compressibility and Mixture Viscosity

A three-dimensional multilayer shallow water approach to study polydisperse sedimentation and sediment transport in a viscous fluid is presented. The fluid is assumed be loaded with finely dispersed solid particles that belong to a finite number of species that differ in density and size. The model formulation allows one to recover the global mass and linear momentum balance laws of the mixture. The model incorporates compressibility of the sediment and viscosity of the mixture through a viscous stress tensor. As a consequence of a dimensional analysis applied to the global mass conservation and linear momentum balance equations, the horizontal components of the compression term and the horizontal terms of the viscous stress tensor may be neglected. This results in a final model that is vertically consistent with the classical one-dimensional vertical model. Numerical simulations illustrate the coupled solids volume fraction and flow fields in various scenarios and the effect of the compressibility and viscosity terms. Various bottom topographies give rise to recirculation of the fluid and high solids volume fractions on the bottom.

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  • Journal IconJournal of Scientific Computing
  • Publication Date IconNov 1, 2020
  • Author Icon Raimund Bürger + 2
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A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented.

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  • Journal IconMathematics
  • Publication Date IconMay 23, 2020
  • Author Icon Ernesto Guerrero Fernández + 2
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Development and Application of a Pre-Corrected Fast Fourier Transform Accelerated Multi-Layer Boundary Element Method for the Simulation of Shallow Water Acoustic Propagation

Because of the complexities associated with the domain geometry and environments, accurate prediction of acoustics propagation and scattering in realistic shallow water environments by direct numerical simulation is challenging. Based on the pre-corrected Fast Fourier Transform (PFFT) method, we accelerated the classical boundary element method (BEM) to predict the acoustic propagation in a multi-layer shallow water environment. The classical boundary element method formulate the acoustics propagation problem as a linear equation system in the form of [A]{x}={b}, where [A] is an N×N dense matrix composed of influence coefficients. Solving such linear equation system requires O(N2/N3) computational cost for iterative/direct methods. The developed method, PFFT-BEM, can effectively reduce the computational efforts for direct numerical simulations from O(N2~3) to O(Nlog N), where N is the total number of boundary unknowns. To numerically simulate the sound propagation in a shallow water environment, we applied the first-order non-reflecting boundary condition in the truncated numerical domain boundary to eliminate the errors due to reflected waves. Multi-layer coupled formulation was used to include the environment inhomogeneity in PFFT-BEM. Through multiple convergence tests on the number of layers and elements, we validated and quantified the accuracy of PFFT-BEM. To demonstrate the usefulness and capability of the developed PFFT-BEM, we simulated three-dimensional (3D) underwater sound propagation through 3D geometries to check the efficacy of the established classical method: the 3D Parabolic equation model. Finally, PFFT-BEM was employed to simulate sound propagation through a complex multi-layer shallow water environment with internal waves. The “3D+T” results obtained by PFFT-BEM compared well with the physical test, thereby proving the capability and correctness of this method.

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  • Journal IconApplied Sciences
  • Publication Date IconApr 1, 2020
  • Author Icon Chengxi Li + 1
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Nonlinear Internal Waves in Multilayer Shallow Water

Nonlinear Internal Waves in Multilayer Shallow Water

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  • Journal IconПрикладная механика и техническая физика
  • Publication Date IconJan 1, 2020
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Nonlinear Internal Waves in Multilayer Shallow Water

Within the framework of the second approximation of the shallow water theory, the flow of a multilayer fluid stratified in density is under study. A mathematical model for the propagation of near-bottom and near-surface large-amplitude internal waves is constructed, taking into account the influence of the fine structure of thermocline (pycnocline). Using the resulting solutions describing the propagation of solitary waves and wave bores, field data are interpreted.

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  • Journal IconJournal of Applied Mechanics and Technical Physics
  • Publication Date IconJan 1, 2020
  • Author Icon V Yu Liapidevskii + 3
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