Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation

Gerbeau, Jean-Frédéric; Perthame, Benôıt

doi:10.3934/dcdsb.2001.1.89

Cited by 284 publications

(271 citation statements)

References 8 publications

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“…Illustration of the notations: (left) interpretation of the unknowns in the vertical plan; (right) finite volume discretization in the horizontal plan A fine derivation from the Navier-Stokes equations is carried out in the work of Gerbeau and Perthame. 19 It is shown that (SW) is an approximation of a sufficiently smooth solution of Navier-Stokes equations in O( ) where is the ratio of the vertical characteristic length to the horizontal characteristic length. A higher-order approximation can be derived but it introduces dissipative terms.…”

Section: Figurementioning

confidence: 99%

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Parisot

2019

Numerical Methods in Fluids

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Summary This work is devoted to the numerical resolution in the multidimensional framework of a hierarchy of reduced models of the water wave equations, such as the Serre‐Green‐Naghdi model. A particular attention is paid to the dissipation of mechanical energy at the discrete level, which acts as a stability argument of the scheme, even with source terms such space and time variation of the bathymetry. In addition, the analysis leads to a natural way to deal with dry areas without leakage of energy. To illustrate the accuracy and the robustness of the strategy, several numerical experiments are carried out. In particular, the strategy is capable of treating dry areas without special treatment.

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Section: Figurementioning

confidence: 99%

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Parisot

2019

Numerical Methods in Fluids

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“…[7] Concerning the coefficient g which appears in the equations (2) and (3), we propose a relationship similar to that used in the viscous regime [Gerbeau and Perthame, 2001;Ferrari and Saleri, 2004]…”

Section: Model Descriptionmentioning

confidence: 99%

Numerical simulation of lava flows based on depth‐averaged equations

Costa

2005

Geophysical Research Letters

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[1] Risks and damages associated with lava flows propagation (for instance the most recent Etna eruptions) require a quantitative description of this phenomenon and a reliable forecasting of lava flow paths. Due to the high complexity of these processes, numerical solution of the complete conservation equations for real lava flows is often practically impossible. To overcome the computational difficulties, simplified models are usually adopted, including 1-D models and cellular automata. In this work we propose a simplified 2D model based on the conservation equations for lava thickness and depthaveraged velocities and temperature which result in first order partial differential equations. The proposed approach represents a good compromise between the full 3-D description and the need to decrease the computational time. The method was satisfactorily applied to reproduce some analytical solutions and to simulate a real lava flow event occurred during the 1991 -93 Etna eruption.Citation: Costa, A., and G. Macedonio (2005), Numerical simulation of lava flows based on depth-averaged equations, Geophys. Res. Lett., 32, L05304,

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“…There are many ways to model friction terms, e.g. [18,19]. In this paper, we focus on the classical Manning formulation (e.g., [20][21][22][23]):…”

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Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

Chertock

Cui

Kurganov

et al. 2015

Numerical Methods in Fluids

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Shallow water models are widely used to describe and study free-surface water flow. Even though, in many practical applications the bottom friction does not have much influence on the solutions, the friction terms will play a significant role when the depth of the water is very small. In this paper, we study the Saint-Venant system of shallow water equations with friction terms and develop a well-balanced central-upwind scheme that is capable of exactly preserving its steady states. The scheme also preserves the positivity of the water depth. We test the designed scheme on a number of one-and two-dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [L. Cea, M. Garrido, and J. Puertas, J. Hydrol, 382 (2010), pp. 88-102], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results.

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Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation

Cited by 284 publications

References 8 publications

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Numerical simulation of lava flows based on depth‐averaged equations

Well‐balanced positivity preserving central‐upwind scheme for the shallow water system with friction terms

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