Analysis and Approximation of Conservation Laws with Source Terms

Greenberg, J. M.; Leroux, Alain; Baraille, Rémy; Noussaïr, Ahmed

doi:10.1137/s0036142995286751

Cited by 127 publications

(109 citation statements)

References 6 publications

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“…σ is an eigenvalue of the intermediate matrix and W n j −W n i is an associated eigenvector. As a consequence, the solution of the linear Riemann problem (14) corresponding to the intercell E ij coincides with the solution of the Riemann problem (10). Nevertheless, the construction of a Roe scheme with Ψ = Φ can be difficult or very costly in practice.…”

Section: Roe Methods For Two-dimensional Nonconservative Systemsmentioning

confidence: 99%

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High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

et al. 2008

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This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in [3] to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.

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Section: Roe Methods For Two-dimensional Nonconservative Systemsmentioning

confidence: 99%

“…In effect, adding to (1) the equation ∂H ∂t = 0, the system can be rewritten in this form (see [12], [6], [7], [9], [10]). The nonconservative products involved in (2) do not make sense in general within the framework of distributions.…”

Section: Introductionmentioning

confidence: 99%

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

et al. 2008

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“…Following an idea introduced in [17,19,20], a system of conservation laws with source terms or balance law ∂W ∂t + ∂F ∂x (W, σ) = S(W, σ) dσ dx , (1.2) σ(x) being a known function, can be considered as a particular case of (1.1), if the trivial equation:…”

Section: Introductionmentioning

confidence: 99%

On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

Parés¹,

Castro²

2004

ESAIM: M2AN

183

213

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Abstract. This paper is concerned with the numerical approximations of Cauchy problems for onedimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of wellbalancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992)

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“…Despite this, numerical schemes based upon a true Godunov-type strategy like for instance Greenberg and LeRoux's method [31], [32] [17] made up of the following steps: • Piecewise constant discretization of variables within each cell.…”

Section: The Convergence Stagnation Problemmentioning

confidence: 99%

“…In particular, one must mention the class of well-balanced schemes, designed to respect steady states. These schemes have been extensively studied by Le Roux and co-workers [28], [31], [32] in the scalar case, and in the framework of the SWE with topography in [11], [12], [13], [44] and with friction in [16]. Zhou and al.…”

Section: Introductionmentioning

confidence: 99%

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

Sahmim¹,

Benkhaldoun

Alcrudo

2007

Journal of Computational Physics

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To cite this version:Slah Sahmim, Fayssal Benkhaldoun, Francisco Alcrudo. A sign matrix based scheme for nonhomogeneous PDE's with an analysis of the convergence stagnation phenomenon. Journal of Computational Physics, Elsevier, 2007, 226 (2) AbstractThis work is devoted to the analysis of a finite volume method recently proposed for the numerical computation of a class of non homogenous systems of partial differencial equations of interest in fluid dynamics. The stability analysis of the proposed scheme leads to the introduction of the sign matrix of the flux jacobian. It appears that this formulation is equivalent to the VFRoe scheme introduced in the homogeneous case and has a natural extension here to non homogeneous systems. Comparative numerical experiments for the Shallow Water and Euler equations with source terms, and a model problem of two phase flow (Ransom faucet) are presented to validate the scheme. The numerical results present a convergence stagnation phenomenon for certain forms of the source term, notably when it is singular. Convergence stagnation has been also shown in the past for other numerical schemes. This issue is addressed in a specific section where an explanation is given with the help of a linear model equation, and a cure is demonstrated.

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Analysis and Approximation of Conservation Laws with Source Terms

Cited by 127 publications

References 6 publications

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

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