Admissibility conditions for Riemann data in shallow water theory

AbstractConsideration is given to the shallow-water equations, a hyperbolic system modeling the propagation of long waves at the surface of an incompressible inviscible fluid of constant depth. It is well known that the solution of the Riemann problem associated to this system may feature dry states for some configurations of the Riemann data. This article will discuss various scenarios in which the Riemann problem for the shallow water system arises in a physically reasonable sense. In particular, it will be shown that if certain physical assumptions on the disposition of the Riemann data are made, then dry states can be avoided in the solution of the Riemann problem.

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Constructive Approach of the Solution of Riemann Problem for Shallow Water Equations with Topography and Vegetation

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2020-0021 ◽

2020 ◽

Vol 28 (2) ◽

pp. 93-114

Author(s):

Stelian Ion ◽

Stefan-Gicu Cruceanu ◽

Dorin Marinescu

Keyword(s):

Initial Data ◽

Shallow Water ◽

Global Solution ◽

Riemann Problem ◽

Hyperbolic System ◽

Shallow Water Equations ◽

Local Existence ◽

Water Model ◽

Constructive Approach ◽

Shallow Water Model

AbstractWe investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not supposed to be “small enough”). One di culty for the extended solution arises from the double degeneracy of the hyperbolic system describing the model. Another di culty is given by the fact that the construction of the solution assumes solving an equation which has no global solution. Finally, we present some cases to illustrate the existence and non-existence of the solution.

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THREE-LAYER APPROXIMATION OF TWO-LAYER SHALLOW WATER EQUATIONS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2013.869269 ◽

2013 ◽

Vol 18 (5) ◽

pp. 675-693 ◽

Cited By ~ 11

Author(s):

Alina Chertock ◽

Alexander Kurganov ◽

Zhuolin Qu ◽

Tong Wu

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Bottom Topography ◽

Water System ◽

Single Layer ◽

Inviscid Fluid ◽

Layer Model ◽

Momentum Exchange ◽

Small Depth ◽

Shallow Water System

Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.

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On the Shallow Water Equations

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0146 ◽

2017 ◽

Vol 72 (9) ◽

pp. 873-879 ◽

Cited By ~ 16

Author(s):

Mahmoud A.E. Abdelrahman

Keyword(s):

Conservation Laws ◽

Shallow Water ◽

Shallow Water Equations ◽

Water System ◽

Diagonal Form ◽

Riemann Invariants ◽

Interesting Application ◽

Nonlinear Conservation Laws ◽

Shallow Water System ◽

Elementary Waves

AbstractWe studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

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Boundary Conditions for Limited Area Models Based on the Shallow Water Equations

Communications in Computational Physics ◽

10.4208/cicp.070312.061112a ◽

2013 ◽

Vol 14 (3) ◽

pp. 664-702 ◽

Cited By ~ 5

Author(s):

Arthur Bousquet ◽

Madalina Petcu ◽

Ming-Cheng Shiue ◽

Roger Temam ◽

Joseph Tribbia

Keyword(s):

Boundary Conditions ◽

Numerical Simulations ◽

Shallow Water ◽

Shallow Water Equations ◽

Water System ◽

Limited Area ◽

Two Dimensional ◽

One Dimensional ◽

Shallow Water System ◽

The Waves

AbstractA new set of boundary conditions has been derived by rigorous methods for the shallow water equations in a limited domain. The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions. The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain. The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle.

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Extending Square Conservation to Arbitrarily Structured C-grids with Shallow Water Equations

10.5194/gmd-2019-122 ◽

2019 ◽

Author(s):

Lilong Zhou ◽

Jinming Feng ◽

Lijuan Hua

Keyword(s):

Total Energy ◽

Shallow Water ◽

Shallow Water Equations ◽

Temporal Integration ◽

Water System ◽

Discrete Operator ◽

Absolute Vorticity ◽

Integral Invariants ◽

Uniform Grids ◽

Shallow Water System

Abstract. The square conservation theory is widely used on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass, total absolute vorticity and instantaneous total energy. The first 2 integral invariants are entirely conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge-Kutta. The method of strictly conserving the total energy simultaneously uses both an anti-symmetrical spatial discrete operator and square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK, attempting to extend the square conservation theory to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations by Institute of Atmospheric Physics (IAP) transformation, expressing the temporal trend of the evolution variable by using the original operators of TRiSK. Using the square conservative Runge-Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.

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Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

Journal of Computational Physics ◽

10.1016/j.jcp.2015.07.043 ◽

2015 ◽

Vol 300 ◽

pp. 202-218 ◽

Cited By ~ 12

Author(s):

Xin Liu ◽

Abdolmajid Mohammadian ◽

Alexander Kurganov ◽

Julio Angel Infante Sedano

Keyword(s):

Shallow Water ◽

System Modeling ◽

Water System ◽

Upwind Scheme ◽

Shallow Water System ◽

Fully Coupled

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Exploring a Multiresolution Modeling Approach within the Shallow-Water Equations

Monthly Weather Review ◽

10.1175/mwr-d-10-05049.1 ◽

2011 ◽

Vol 139 (11) ◽

pp. 3348-3368 ◽

Cited By ~ 73

Author(s):

Todd D. Ringler ◽

Doug Jacobsen ◽

Max Gunzburger ◽

Lili Ju ◽

Michael Duda ◽

...

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Truncation Error ◽

Water System ◽

Test Cases ◽

Grid Spacing ◽

Variable Resolution ◽

Mesh Resolution ◽

Water Test ◽

Shallow Water System

Abstract The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While the long-term motivation for this study is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, the process begins with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of the approach developed herein. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine- and coarse-mesh resolution. The shallow-water system is explored with a suite of meshes ranging from quasi-uniform resolution meshes, where the grid spacing is globally uniform, to highly variable resolution meshes, where the grid spacing varies by a factor of 16 between the fine and coarse regions. The potential vorticity is found to be conserved to within machine precision and the total available energy is conserved to within a time-truncation error. This result holds for the full suite of meshes, ranging from quasi-uniform resolution and highly variable resolution meshes. Based on shallow-water test cases 2 and 5, the primary conclusion of this study is that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This conclusion is consistent with results obtained by others. When these variable-resolution meshes are used for the simulation of an unstable zonal jet, the core features of the growing instability are found to be largely unchanged as the variation in the mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multiresolution climate system modeling.

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