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

Berthon, Christophe; Chalons, Christophe

doi:10.1090/mcom3045

Cited by 56 publications

(51 citation statements)

References 42 publications

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“…More recently, in [20,21], the authors have proposed an extension of the work by Gosse [3] in order to deal with Godunovtype schemes. Such Godunov-type schemes (see [22,23]) are based on approximate Riemann solvers, whose intermediate…”

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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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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“…Note that the left and right states of this dam-break are moving steady states, which satisfy the equation (2). As a consequence, they will be exactly preserved by the firstorder well-balanced scheme: the goal of this experiment is to display the accuracy gained by the use of the θ-WB scheme.…”

Section: Dam-break Experimentsmentioning

confidence: 98%

“…Note that, for a steady state, we have Φ = cst as well as q = cst, as per (2). Let us then define the spatial errors to a steady state, as follows: These remarks lead us to consider switching between the well-balanced and the MUSCL scheme when the time update of the MUCSL scheme becomes very small.…”

Section: A Second-order Accurate Convex Combinationmentioning

confidence: 99%

“…The boundary conditions, q(t, 0) = q 0 and h(t, 25) = h 0 (with q 0 = 4.42), ensure that the solution is a transient state followed by a smooth steady state with nonzero velocity. This steady state is governed by (2).…”

Section: Well-balance Of the Scheme: Capture Of A Steady Statementioning

confidence: 99%

“…schemes that exactly preserve such steady solutions, have been derived in the last decade (see for instance [2,4,5,8]). Namely, in [8], the authors suggested a well-balanced Godunov-type scheme based on a two-state approximate Riemann solver.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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

Cited by 56 publications

References 42 publications

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

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

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

Finite Volume Methods: Foundation and Analysis

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