scholarly journals An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

2021 ◽  
Vol 90 (1) ◽  
Author(s):  
E. Guerrero Fernández ◽  
M. J. Castro Díaz ◽  
M. Dumbser ◽  
T. Morales de Luna
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Stationary Solutions ◽  
Variable Density ◽  
High Order ◽  
Water Model ◽  
Variable Pressure ◽  
Shallow Water Model ◽  
Hyperbolic Pde ◽  
Coarse Meshes

AbstractIn 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.

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 Eastward-moving convection-enhanced modons in shallow water in the equatorial tangent plane

2019 ◽  
Author(s):  
M. Rostami
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Large Scale ◽  
Tangent Plane ◽  
Water Model ◽  
Moist Convection ◽  
Shallow Water Model ◽  
Beta Plane ◽  
Vapour Condensation ◽  
Velocity Approximation

We report a discovery of steady long-living slowly eastward moving large-scale coherent twin cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing analytical asymptotic modon solutions in the non-divergent velocity approximation and then show by simulations with a high-resolution numerical scheme that such configurations evolve into steady dipolar solutions of the full model. In the atmospheric context, the modons persist in the presence of moist convection, being accompanied and enhanced by specific patterns of water-vapour condensation.

scholarly journals A high-order conservative collocation scheme and its application to global shallow water equations

2014 ◽  
Vol 7 (4) ◽  
pp. 4251-4290 ◽  
Author(s):  
C. Chen ◽  
X. Li ◽  
X. Shen ◽  
F. Xiao
Keyword(s):  
Shallow Water ◽  
General Circulation ◽  
Control Volume ◽  
High Order ◽  
High Order Accuracy ◽  
Water Model ◽  
Shallow Water Model ◽  
Order Accuracy ◽  
Flux Function ◽  
Dependent Variables

Abstract. An efficient and conservative collocation method is proposed and used to develop a global shallow water model in this paper. Being a nodal type high-order scheme, the present method solves the point-wise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve the high-order accuracy. The time evolution equations to update the unknowns are derived under the flux-reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the point-wise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the control volume which are obtained as the Riemann solutions between the adjacent cells. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

scholarly journals A Parallel High-Order Discontinuous Galerkin Shallow Water Model

2009 ◽  
pp. 63-72 ◽  
Author(s):  
Claes Eskilsson ◽  
Yaakoub El-Khamra ◽  
David Rideout ◽  
Gabrielle Allen ◽  
Q. Jim Chen ◽  
...  
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
High Order ◽  
Water Model ◽  
Shallow Water Model

A high-order fully explicit flux-form semi-Lagrangian shallow-water model

2014 ◽  
Vol 75 (2) ◽  
pp. 103-133 ◽  
Author(s):  
Paul Aaron Ullrich ◽  
Peter Hjort Lauritzen ◽  
Christiane Jablonowski
Keyword(s):  
Shallow Water ◽  
High Order ◽  
Water Model ◽  
Shallow Water Model
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