Shallow Water Flows in Channels

Hernández-Dueñas, Gerardo; Karni, Smadar

doi:10.1007/s10915-010-9430-x

Cited by 28 publications

(25 citation statements)

References 23 publications

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“…We compare the numerical results obtained by the present scheme (dotted line) to the numerical solution obtained using the upwind Roe-type scheme (solid line) in [14]. The comparison shows a good agreement between the two schemes.…”

Section: Perturbations Of Steady State Of Restmentioning

confidence: 82%

“…The shallow water equations for flows through channels with variable cross-section are given by, [14],…”

Section: The Modelmentioning

confidence: 99%

“…in the second component of the flux f in (14). In order to prevent this, after reconstructing Q, w, and A T at the cell interfaces, we use the regularization technique suggested by [18],…”

Section: Regularization Of Flow Velocity and Discharge For Small Amentioning

confidence: 99%

“…The proposed numerical scheme preserves steady states at rest by construction. It has been shown in related works (e.g., [11,14,28]) that recognizing steady states at rest is enough to enable the scheme to recognize and compute near steady state solutions accurately. We begin testing our numerical scheme with the evolution of a perturbation from a steady state.…”

Section: Perturbations Of Steady State Of Restmentioning

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: Perturbations Of Steady State Of Restmentioning

confidence: 82%

“…The shallow water equations for flows through channels with variable cross-section are given by, [14],…”

Section: The Modelmentioning

confidence: 99%

“…in the second component of the flux f in (14). In order to prevent this, after reconstructing Q, w, and A T at the cell interfaces, we use the regularization technique suggested by [18],…”

Section: Regularization Of Flow Velocity and Discharge For Small Amentioning

confidence: 99%

Section: Perturbations Of Steady State Of Restmentioning

confidence: 99%

See 2 more Smart Citations

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

Balbás

Hernández-Dueñas

2014

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…Many upwind (see, e.g., [1,2,5,9,10,18,22,31,33,37] ) and central (see, e.g., [6,8,15,23,28,40,41,45] ) schemes for the shallow water system (1) , which is a hyperbolic system of conservation (if B x ≡ B y ≡ 0) or balance (if B is not a constant) laws, have been proposed in the past two decades. Roughly speaking, the main difference between upwind and central schemes is that upwind schemes use characteristic information and utilize (approximate) Riemann problem solvers to determine nonlinear wave propagation, while central schemes are based on averaging over the waves without using their detailed structures.…”

Section: Introductionmentioning

confidence: 99%

Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows

2016

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a b s t r a c tWe develop a new second-order two-dimensional central-upwind scheme on cell-vertex grids for approximating solutions of the Saint-Venant system with source terms due to bottom topography. Centralupwind schemes are developed based on the information about the local speeds of wave propagation. Compared to the triangular central-upwind schemes, the proposed cell-vertex one has an advantage of using more cell interfaces which provide more information on the waves propagating in different directions. We propose a new piecewise linear approximation of the bottom topography and a novel nonoscillatory reconstruction in which the gradient of each variable is computed using a modified minmodtype method to ensure the stability of the scheme. A new technique is proposed for the correction of the water surface elevation which guarantees the positivity of the water depth. The well-balanced property of the proposed central-upwind scheme is ensured using a special discretization for the cell averages of the topography source terms. The proposed scheme is tested on a number of numerical examples, among which we consider steady-state solutions with almost dry areas and their perturbations and solutions with rapidly varying flows over discontinuous bottom topography. Our numerical experiments confirm stability, well-balanced, positivity preserving properties and second-order accuracy of the proposed method. This scheme can be applied to shallow water models when the bed topography is discontinuous and/or highly oscillatory, and on complicated domains where the use of unstructured grids is advantageous.

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Exactly well‐balanced positivity preserving nonstaggered central scheme for open‐channel flows

Dong

2020

Numerical Methods in Fluids

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In this paper, we construct and study an exactly well-balanced positivity-preserving nonstaggered central scheme for shallow water flows in open channels with irregular geometry and nonflat bottom topography. We introduce a novel discretization of the source term based on hydrostatic reconstruction to obtain the exactly well-balanced property for the still water steady-state solution even in the presence of wetting and drying transitions. The positivity-preserving property of the cross-sectional wet area is obtained by using a modified "draining" time-step technique. The current scheme is also Riemann-solver-free. Several classical problems of open-channel flows are used to test these properties. Numerical results confirm that the current scheme is robust, exactly well-balanced and positivity-preserving.

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Shallow Water Flows in Channels

Cited by 28 publications

References 23 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

Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows

Exactly well‐balanced positivity preserving nonstaggered central scheme for open‐channel flows

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