scholarly journals A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 848
Author(s):  
Ernesto Guerrero Fernández ◽  
Manuel Jesús Castro-Díaz ◽  
Tomás Morales de Luna
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Hyperbolic Equations ◽  
Laboratory Data ◽  
Variable Density ◽  
Second Order ◽  
Water Model ◽  
Finite Volume Scheme ◽  
Reconstruction Technique ◽  
Shallow Water Model

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.

scholarly journals A Finite-Volume Icosahedral Shallow-Water Model on a Local Coordinate

2009 ◽  
Vol 137 (4) ◽  
pp. 1422-1437 ◽  
Author(s):  
Jin-Luen Lee ◽  
Alexander E. MacDonald
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Spherical Surface ◽  
Second Order ◽  
Water Model ◽  
Numerical Dissipation ◽  
Shallow Water Model ◽  
Finite Volume Model ◽  
Volume Model ◽  
Cartesian Coordinate

Abstract An icosahedral-hexagonal shallow-water model (SWM) on the sphere is formulated on a local Cartesian coordinate based on the general stereographic projection plane. It is discretized with the third-order Adam–Bashforth time-differencing scheme and the second-order finite-volume operators for spatial derivative terms. The finite-volume operators are applied to the model variables defined on the nonstaggered grid with the edge variables interpolated using polynomial interpolation. The projected local coordinate reduces the solution space from the three-dimensional, curved, spherical surface to the two-dimensional plane and thus reduces the number of complete sets of basis functions in the Vandermonde matrix, which is the essential component of the interpolation. The use of a local Cartesian coordinate also greatly simplifies the mathematic formulation of the finite-volume operators and leads to the finite-volume integration along straight lines on the plane, rather than along curved lines on the spherical surface. The SWM is evaluated with the standard test cases of Williamson et al. Numerical results show that the icosahedral SWM is free from Pole problems. The SWM is a second-order finite-volume model as shown by the truncation error convergence test. The lee-wave numerical solutions are compared and found to be very similar to the solutions shown in other SWMs. The SWM is stably integrated for several weeks without numerical dissipation using the wavenumber 4 Rossby–Haurwitz solution as an initial condition. It is also shown that the icosahedral SWM achieves mass conservation within round-off errors as one would expect from a finite-volume model.

scholarly journals Flood Simulation by a Well-Balanced Finite Volume Method in Tapi River Basin, Thailand, 2017

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Sutatip Vichiantong ◽  
Thida Pongsanguansin ◽  
Montri Maleewong
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Water Model ◽  
Interface Problem ◽  
Reconstruction Technique ◽  
Shallow Water Model ◽  
Flood Simulation ◽  
Southern Thailand ◽  
Volume Method

Flood simulation of a region in southern Thailand during January 2017 is presented in this work. The study area covers the Tapi river, the longest river in southern Thailand. The simulation is performed by applying the two-dimensional shallow water model in the presence of strong source terms to the local bottom topography. The model is solved numerically by our finite volume method with well-balanced property and linear reconstruction technique. This technique is accurate and efficient at solving for complex flows in the wet/dry interface problem. Measurements of flows are collected from two gauging stations in the area. The initial conditions are prepared to match the simulated flow to the measurements recorded at the gauging stations. The accuracy of the numerical simulations is demonstrated by comparing the simulated flood area to satellite images from the same period. The results are in good agreement, indicating the suitability of the shallow water model and the presented numerical method for simulating floodplain inundation.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

scholarly journals A Well-balanced Finite Volume Scheme for Shallow Water Equations with Porosity: Application to Modelling Flow through Rigid Vegetation

2018 ◽  
Vol 40 ◽  
pp. 05032
Author(s):  
Minh H. Le ◽  
Virgile Dubos ◽  
Marina Oukacine ◽  
Nicole Goutal
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Simple Wave ◽  
Water Model ◽  
Strong Interactions ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Rigid Vegetation ◽  
Vertical Cylinders

Strong interactions exist between flow dynamics and vegetation in open-channel. Depth-averaged shallow water equations can be used for such a study. However, explicit representation of vegetation can lead to very high resolution of the mesh since the vegetation is often modelled as vertical cylinders. Our work aims to study the ability of a single porosity-based shallow water model for these applications. More attention on flux and source terms discretizations are required in order to archive the well-balancing and shock capturing properties. We present a new Godunov-type finite volume scheme based on a simple-wave approximation and compare it with some other methods in the literature. A first application with experimental data was performed.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

A dynamic multilayer shallow water model for polydisperse sedimentation

2019 ◽  
Vol 53 (4) ◽  
pp. 1391-1432 ◽  
Author(s):  
Raimund Bürger ◽  
Enrique D. Fernández-Nieto ◽  
Víctor Osores
Keyword(s):  
Shallow Water ◽  
Bottom Topography ◽  
Mining Industry ◽  
Mass Conservation ◽  
Variable Density ◽  
Solid Particles ◽  
Water Model ◽  
Reduced Form ◽  
Momentum Balance ◽  
Shallow Water Model

A multilayer shallow water approach for the approximate description of polydisperse sedimentation in a viscous fluid is presented. The fluid is assumed to carry finely dispersed solid particles that belong to a finite number of species that differ in density and size. These species segregate and form areas of different composition. In addition, the settling of particles influences the motion of the ambient fluid. A distinct feature of the new approach is the particular definition of the average velocity of the mixture. It takes into account the densities of the solid particles and the fluid and allows us to recover the global mass conservation and linear momentum balance laws of the mixture. This definition motivates a modification of the Masliyah–Lockett–Bassoon (MLB) settling velocities of each species. The multilayer shallow water model allows one to determine the spatial distribution of the solid particles, the velocity field, and the evolution of the free surface of the mixture. The final model can be written as a multilayer model with variable density where the unknowns are the average velocities and concentrations in each layer, the transfer terms across each interface, and the total mass. An explicit formula of the transfer terms leads to a reduced form of the system. Finally, an explicit bound of the minimum and maximum eigenvalues of the transport matrix of the system is utilized to design a Harten–Lax–van Leer (HLL)-type path-conservative numerical method. Numerical simulations illustrate the coupled polydisperse sedimentation and flow fields in various scenarios, including sedimentation in a type of basin that is used in practice in mining industry and in a basin whose bottom topography gives rise to recirculations of the fluid and high solids concentrations.

scholarly journals Topography based local spherical Voronoi grid refinement on classical and moist shallow-water finite volume models

2021 ◽  
Author(s):  
Luan F. Santos ◽  
Pedro S. Peixoto
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Computational Cost ◽  
Water Model ◽  
Uniform Grid ◽  
Small Scale ◽  
Finite Volume Scheme ◽  
Local Refinement ◽  
The Andes ◽  
Smooth Transitions

Abstract. Locally refined grids for global atmospheric models are attractive since they are expected to provide an alternative to solve local phenomena without the requirement of a global high-resolution uniform grid, whose computational cost may be prohibitive. The Spherical Centroidal Voronoi Tesselations (SCVT), as used in the atmospheric Model for Prediction Across Scales (MPAS), allows a flexible way to build and work with local refinement. Alongside, the Andes Range plays a key role in the South American weather, but it is hard to capture its fine structure dynamics in global models. This paper describes how to generate SCVT grids that are locally refined in South America and that also capture the sharp topography of the Andes Range by defining a density function based on topography and smoothing techniques. We investigate the use of the mimetic finite volume scheme employed in the MPAS dynamical core on this grid considering the non-linear classic and moist shallow-water equations on the sphere. We show that the local refinement, even with very smooth transitions from different resolutions, generates spurious numerical inertia-gravity waves that may even numerically de-stabilize the model. In the moist shallow-water model, where physical processes such as precipitation and cloud formation are included, our results show that the local refinement may generate spurious rain that is not observed in uniform resolution SCVT grids. Fortunately, the spurious waves originate from small-scale grid-related numerical errors and therefore can be mitigated using small amounts of numerical diffusion. We show that, in some cases, the clouds are better represented in a variable resolution grid when compared to a respective uniform resolution grid with the same number of cells, while in other cases, grid effects can deteriorate the cloud and rain representation.

scholarly journals Topography-based local spherical Voronoi grid refinement on classical and moist shallow-water finite-volume models

2021 ◽  
Vol 14 (11) ◽  
pp. 6919-6944
Author(s):  
Luan F. Santos ◽  
Pedro S. Peixoto
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Voronoi Tessellation ◽  
Computational Cost ◽  
Water Model ◽  
Small Scale ◽  
Finite Volume Scheme ◽  
Local Refinement ◽  
The Andes ◽  
Smooth Transitions

Abstract. Locally refined grids for global atmospheric models are attractive since they are expected to provide an alternative to solve local phenomena without the requirement of a global high-resolution uniform grid, whose computational cost may be prohibitive. Spherical centroidal Voronoi tessellation (SCVT), as used in the atmospheric Model for Prediction Across Scales (MPAS), allows a flexible way to build and work with local refinement. In addition, the Andes Range plays a key role in the South American weather, but it is hard to capture its fine-structure dynamics in global models. This paper describes how to generate SCVT grids that are locally refined in South America and that also capture the sharp topography of the Andes Range by defining a density function based on topography and smoothing techniques. We investigate the use of the mimetic finite-volume scheme employed in the MPAS dynamical core on this grid considering the nonlinear classic and moist shallow-water equations on the sphere. We show that the local refinement, even with very smooth transitions from different resolutions, generates spurious numerical inertia–gravity waves that may even numerically destabilize the model. In the moist shallow-water model, wherein physical processes such as precipitation and cloud formation are included, our results show that the local refinement may generate spurious rain that is not observed in uniform-resolution SCVT grids. Fortunately, the spurious waves originate from small-scale grid-related numerical errors and can therefore be mitigated using fourth-order hyperdiffusion. We exploit a grid geometry-based hyperdiffusion that is able to stabilize spurious waves and has very little impact on the total energy conservation. We show that, in some cases, the clouds are better represented in a variable-resolution grid when compared to a respective uniform-resolution grid with the same number of cells, while in other cases, grid effects can affect the cloud and rain representation.

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