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

Michel-Dansac, Victor; Berthon, Christophe; Clain, Stéphane; Foucher, Franoise

doi:10.1016/j.jcp.2017.01.009

Cited by 37 publications

(39 citation statements)

References 48 publications

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“…Namely, in [9], the authors propose a well-balanced scheme for the nonlinear Manning friction source term. Due to the nonlinearity, providing a parameter-free second-order extension of this scheme would be an interesting challenge.…”

Section: Resultsmentioning

confidence: 99%

A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography

Berthon

Loubère

Michel-Dansac

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We consider the well-balanced numerical scheme for the shallow-water equations with topography introduced in [8] and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present contribution is to derive a parameter-free second-order well-balanced scheme. To that end, we consider a convex combination between the well-balanced scheme and a second-order scheme. We then prove that a relevant choice of the parameter of this convex combination ensures that the resulting scheme is both second-order accurate and well-balanced. Afterwards, we perform several numerical experiments, in order to illustrate both the second-order accuracy and the well-balance property of this numerical scheme. Finally, we outline some perspectives in a short conclusion.

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Section: Resultsmentioning

confidence: 99%

A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography

Berthon

Loubère

Michel-Dansac

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…As a consequence, as long as the solution stays far away from dry areas, with h L ≥ h 0 > 0 and h R ≥ h 0 > 0, and with a smooth topography function, bothh L andh R remain positive for small enough ∆x. Moreover, the condition (37) imposes that the reconstructed water heights h − L and h + R remain positive. As a consequence, at the level of the scheme presentation, we assume to be far away from dry areas.…”

Section: Discrete Entropy Inequalitymentioning

confidence: 99%

“…To avoid these difficulties, some works suggest to consider weaker formulations of the entropy stability. For instance, recently in [5,36,37], extensions of the HLL scheme [27] produce entropy consistent schemes (in the sense where entropy inequlity is reached up to O(∆x)) able to exactly capture all the steady states (at rest or moving). In [1], the authors establish an entropy inequality satisfied by the semi-discrete (time continuous) scheme associated with the hydrostatic reconstruction (11).…”

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Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities

et al. 2019

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This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.

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“…However, it is the simplest one. Some works in the literature consider more complex steady states [14,29,38,39].…”

Section: Preliminary Analysis Of the Congested Shallow Water Modelmentioning

confidence: 99%

Congested shallow water model: roof modeling in free surface flow

et al. 2018

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We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

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

Cited by 37 publications

References 48 publications

A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography

A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography

Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities

Congested shallow water model: roof modeling in free surface flow

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