High order well balanced schemes for systems of balance laws

Russo, Giovanni; Khe, A. K.

doi:10.1090/psapm/067.2/2605287

Cited by 21 publications

(16 citation statements)

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“…Since we have imposed a smooth topography, in the present work we deal with smooth steady states. The reader is referred to [13,33,38,39] where extension to discontinuous buttom is considered that makes discontinuous the steady water height (see also [28,29] to related studies).…”

Section: Characteristic Fields and Riemann Invariants Easy Calculatimentioning

confidence: 99%

“…During the last two decades, after the works by Bermudez-Vasquez [2] and Greenberg-LeRoux [24] (see also [21,22,23]), the derivation of well-balanced schemes able to restore the lake at rest (1.4) was a very active research topic. Several strategies have been derived (for instance see [1,30,26,10,21,24,11,32,31,34,33,12,38,39]). The main difficulty coming from the derivation of wellbalanced schemes remains in the discretization of the topography source term to be consistent with the lake at rest.…”

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A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

Berthon

Chalons

2015

Math. Comp.

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This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.

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Section: Characteristic Fields and Riemann Invariants Easy Calculatimentioning

confidence: 99%

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confidence: 99%

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

Berthon

Chalons

2015

Math. Comp.

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“…Next, in [3], a pioneer fully well-balanced scheme was designed to deal with these sensitive steady states, where the numerical technique is based on a suitable resolution of the Bernoulli equation. Next, several methods preserving the moving steady states were designed by involving high-order accurate techniques (see [15][16][17][18] for high-order and exactly well-balanced schemes, and [19] for a high-order accurate scheme on all steady state configurations).…”

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

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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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“…A good numerical method for the system (1.1) should thus be well-balanced (in the sense that it must exactly preserve physically relevant steady states) and positivity preserving (in the sense that the computed values of h must be positive). In the past two decades, many well-balanced schemes have been developed (e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Some of them preserve only 'lake at rest' steady states, that is, u Á v Á 0, h C B Á constant, [2-10, 12, 13], other can preserve a nonflat steady-state solution as well, [11,[15][16][17].…”

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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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High order well balanced schemes for systems of balance laws

Cited by 21 publications

References 0 publications

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

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

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

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