Efficient well-balanced hydrostatic upwind schemes for shallow-water equations

Berthon, Christophe; Foucher, Françoise

doi:10.1016/j.jcp.2012.02.031

Cited by 60 publications

(81 citation statements)

References 46 publications

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“…Some of them have been extensively studied. For instance, by adopting q 0 = 0 and assuming h > 0, we recover the notable lake at rest steady state (see for instance [30,5,27,31,32,2]), defined by  q = 0, h + Z = cst. (2.6) We now turn to the study of (2.5) for q 0 ̸ = 0.…”

Section: Smooth Steady States With Positive Water Heightsmentioning

confidence: 83%

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A well-balanced scheme for the shallow-water equations with topography

Michel-Dansac¹,

Berthon²,

Clain³

et al. 2016

Computers & Mathematics with Applications

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International audienceA non-negativity preserving and well-balanced scheme that exactly preserves all the smooth steady states of the shallow water system, including the moving ones, is proposed. In addition, the scheme must deal with vanishing water heights and transitions between wet and dry areas. A Godunov-type method is derived by using a relevant average of the source terms within the scheme, in order to enforce the required well-balance property. A second-order well-balanced MUSCL extension is also designed. Numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

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Section: Smooth Steady States With Positive Water Heightsmentioning

confidence: 83%

“…Such approaches were extended by Gosse [3] for nonlinear systems, by involving a nonlinear equation to be solved. More recently, in [4], the authors proposed a simplification (by enforcing vanishing velocities) of Gosse's work [3], yielding the so-called hydrostatic reconstruction (see [5][6][7][8][9][10][11][12][13] for related work).…”

Section: Introductionmentioning

confidence: 99%

A well-balanced scheme for the shallow-water equations with topography

Michel-Dansac¹,

Berthon²,

Clain³

et al. 2016

Computers & Mathematics with Applications

Self Cite

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“…For instance, Stelling and Duinmeijer [14] provided a numerical scheme that combined mass conservation with alternatively momentum balance or energy head conservation depending on the flow conditions while recently Fjordholm et al [15] designed energy conservative finite volume schemes using energy stable numerical diffusion operators. Recently, Berthon and Foucher [16] presented a numerical procedure able to modify any finite volume scheme given for the homogeneous shallow-water equations in order to deal with preservation of motionless steady state over irregular geometries.…”

Section: Introductionmentioning

confidence: 99%

Energy balance numerical schemes for shallow water equations with discontinuous topography

Murillo

García‐Navarro

2013

Journal of Computational Physics

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“…We then apply the midpoint rule to approximate the integrals on the right-hand side (RHS) of (14) to obtain the well-balanced quadrature for S (2) j :…”

Section: Well-balanced Discretization Of the Source Termmentioning

confidence: 99%

“…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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Efficient well-balanced hydrostatic upwind schemes for shallow-water equations

Cited by 60 publications

References 46 publications

A well-balanced scheme for the shallow-water equations with topography

A well-balanced scheme for the shallow-water equations with topography

Energy balance numerical schemes for shallow water equations with discontinuous topography

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

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