Central-Upwind Schemes for Two-Layer Shallow Water Equations

Kurganov, Alexander; Petrova, Guergana

doi:10.1137/080719091

Cited by 96 publications

(100 citation statements)

References 31 publications

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“…The scheme extends previous works in [5,17,18,19] to flows along channels with variable geometry. This extension is not trivial; the varying geometry of the channel leads to fluxes and source terms that require the approximation of integral terms, making the balance of them more difficult; while in channels with constant width (σ ≡ 1), well-balancing may be accomplished solely by choosing an appropriate discretization of the source term, in the variable geometry case, the conserved variables A = B+h B…”

Section: Numerical Schemesupporting

confidence: 57%

“…To this end, it is convenient -following [19]-to reformulate (1) in terms of the total elevation of the free water layer, w = h + B and its total area,…”

Section: Numerical Schemementioning

confidence: 99%

“…Satisfying also the well balance property for Q j (t) requires additional considerations; the cell average of the source term, S j , in (15) needs to be discretized so as to balance the numerical fluxes (19). In order to find such discretezation, we start by writing the second component of the numerical flux difference in (18) for the rest conditions (6).…”

Section: Well Balancementioning

confidence: 99%

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A positivity preserving central scheme for shallow water flows in channels with wet-dry states

Balbás

Hernández-Dueñas

2014

ESAIM: M2AN

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We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

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Section: Numerical Schemesupporting

confidence: 57%

“…To this end, it is convenient -following [19]-to reformulate (1) in terms of the total elevation of the free water layer, w = h + B and its total area,…”

Section: Numerical Schemementioning

confidence: 99%

Section: Well Balancementioning

confidence: 99%

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A positivity preserving central scheme for shallow water flows in channels with wet-dry states

Balbás

Hernández-Dueñas

2014

ESAIM: M2AN

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“…For the two-layer shallow water equations of the first class for nonmiscible fluids, the well-established Roe's scheme [25] has applied in [12] among others. Techniques based on central-upwind schemes using the surface elevation instead of the water depth have also been used in [18] for numerical solution of the two-layer shallow water equations for nonmiscible fluids. In [10] numerical methods based on kinetic reconstructions have been studied for these two-layer shallow water equations.…”

Section: Introductionmentioning

confidence: 99%

A fast finite volume solver for multi-layered shallow water flows with mass exchange

Audusse¹,

Benkhaldoun²,

Sari³

et al. 2014

Journal of Computational Physics

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(2014) 'A fast nite volume solver for multi-layered shallow water ows with mass exchange.', Journal of computational physics., 272 . pp. 23-45. Further information on publisher's website:http://dx.doi.org/10.1016/j.jcp.2014.04.026Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Journal of Computational Physics, 272, 2014Physics, 272, , 10.1016Physics, 272, /j.jcp.2014 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractA fast finite volume solver for hydrostatic multi-layered shallow water flows with mass exchange is investigated. In contrast to many models for multi-layered hydrostatic shallow water flows where the immiscible suppression is assumed, the present model allows for mass exchange between the layers. The multi-layered shallow water equations form a system of conservation laws with source terms for which the computation of the eigenvalues is not trivial. For most practical applications, complex eigenvalues may arise in the system and the multi-layered shallow water equations are not hyperbolic any more. This property makes the application of conventional finite volume methods difficult or even impossible for those methods which require in their formulation the explicit computation of the eigenvalues. In the current study, we propose a finite volume method that avoids the solution of Riemann problems. At each time step, the method consists of two stages to update the new solution. In the first stage, the multilayered shallow water equations are rewritten in a non-conservative form and the intermediate solutions are calculated using the method of characteristics. In the second stage, the numerical fluxes are reconstructed from the intermediate solutions in the first stage and used in the conservative form of the multi-layered shallow water equations. The proposed method is simple to implement, satisfies the conservation property and is suitable for multi-layered shallow water equations on non-flat topography. The proposed finite volume solver is verified against several benchmark tests and it shows good agreement with analytical solutions of the incompressible hydrostat...

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“…Each layer is assumed to have a constant density, ρ i , i = 1, 2 (ρ 1 < ρ 2 ). The unknowns q i (x, t) and h i (x, t) represent respectively the mass-flow and the thickness of the ith layer at the section of coordinate x at time t. The numerical resolution of two-layer or multilayer shallow water systems has been object of an intense research during the last years: see for instance [1], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [22], [24] . .…”

Section: Introductionmentioning

confidence: 99%

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

2011

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The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see [14]). This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞ -stable, well-balanced, and it doesn't require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

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Central-Upwind Schemes for Two-Layer Shallow Water Equations

Cited by 96 publications

References 31 publications

A positivity preserving central scheme for shallow water flows in channels with wet-dry states

A positivity preserving central scheme for shallow water flows in channels with wet-dry states

A fast finite volume solver for multi-layered shallow water flows with mass exchange

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

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