A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations

Greenberg, J. M.; Leroux, Alain

doi:10.1137/0733001

Cited by 602 publications

(460 citation statements)

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“…Recently, Greenberg and LeRoux have proposed to treat the particular case g(u) = −u in a different way inside a Godunov type scheme, [23]. One motivation was to get rid of the error coming from the projection stage lying between the treatment of the convective and the reactive part.…”

Section: Lemma 6 Under the Assumptions Of Lemma 5 The Relaxation Esmentioning

confidence: 99%

“…Its entropy solution is made of two elementary waves; we refer to [19,23] for more details in this direction. The general results of [12,32,36] guarantee uniqueness for these Riemann problems, but the next lemma establishes uniqueness for the Cauchy problem (49) under some restriction on the right-hand side.…”

Section: Lemmamentioning

confidence: 99%

“…We first recall the function φ from the very early papers, [23,22,17] (see also the standing wave curves (11), (12) in [27]):…”

Section: Lemmamentioning

confidence: 99%

“…For our examples, we distinguished between relaxation, for which the convergence towards the equilibrium manifold (the local Maxwellian) is the key point and the general balance law, where a longtime decay to some steady regime should occur in order to respect the dynamics of the differential underlying problem, [35]. It turns out that these two criteria are actually difficult to meet within the existing approaches, [7,8,13,23].…”

Section: Introductionmentioning

confidence: 99%

“…Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in [5,9,12,24,32,36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17,19,20,22,23]; see also [8,21,40,39] for closely related works. We stress that our results do not contradict [26].…”

Section: Introductionmentioning

confidence: 99%

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Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

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Section: Lemma 6 Under the Assumptions Of Lemma 5 The Relaxation Esmentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

“…We first recall the function φ from the very early papers, [23,22,17] (see also the standing wave curves (11), (12) in [27]):…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

Cordier

Lucas

2015

Water Resources Research

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Overland flow and soil erosion play an essential role in water quality and soil degradation.Such processes, involving the interactions between water flow and the bed sediment, are classically described by a well-established system coupling the shallow water equations and the Hairsine-Rose model. Numerical approximation of this coupled system requires advanced methods to preserve some important physical and mathematical properties; in particular, the steady states and the positivity of both water depth and sediment concentration. Recently, finite volume schemes based on Roe's solver have been proposed by Heng et al. (2009) and Kim et al. (2013) for one and two-dimensional problems. In their approach, an additional and artificial restriction on the time step is required to guarantee the positivity of sediment concentration. This artificial condition can lead the computation to be costly when dealing with very shallow flow and wet/dry fronts. The main result of this paper is to propose a new and faster scheme for which only the CFL condition of the shallow water equations is sufficient to preserve the positivity of sediment concentration. In addition, the numerical procedure of the erosion part can be used with any well-balanced and positivity preserving scheme of the shallow water equations. The proposed method is tested on classical benchmarks and also on a realistic configuration.

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A well‐balanced FV scheme for compound channels with complex geometry and movable bed

Minatti

2015

Water Resources Research

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This work focuses on the implementation of a Shallow Water-Exner model for compound natural channels with complex geometry and movable bed within the finite volume framework. The model is devised for compound channels modeling: cross-section overbanks are treated with fixed bed conditions, while the main channel is left free to modify its morphology. A capacitive approach is used for bed load transport modeling, in which the solid flow rates are estimated with bed load transport formulas. The model equations pose some numerical issues in the case of natural channels, where bed load transport may occur for both subcritical and supercritical flows and geometry varies in space. An explicit path-conservative scheme, designed to overcome all these issues, is presented in the paper. The scheme solves liquid and solid phases dynamics in a coupled manner, in order to correctly model near critical currents/channel interactions and is well-balanced, that is able to properly reproduce steady states. The Roe and Osher Riemann solvers are implemented, so as to take into account the spatial geometry variations of natural channels. The scheme reaches up to second-order accuracy. Validation is performed with fixed and movable bed test cases whose analytical solution is known, and with flume experimental data. An application of the model to a real case study is also shown.

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A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations

Cited by 602 publications

References 4 publications

Localization effects and measure source terms in numerical schemes for balance laws

Localization effects and measure source terms in numerical schemes for balance laws

A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

A well‐balanced FV scheme for compound channels with complex geometry and movable bed

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