Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

Martin, Sébastien; Vovelle, Julien

doi:10.1090/s0033-569x-07-01061-7

Cited by 2 publications

(3 citation statements)

References 27 publications

(21 reference statements)

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“…However, similarly to the results obtained in [17] for regular fluxes, the large-time behaviour of the entropy solution highly depends on the initial condition.…”

Section: Numerical Resultssupporting

confidence: 73%

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Martin

Vovelle²

2008

ESAIM: M2AN

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Abstract. This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687-705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).Mathematics Subject Classification. 35L65, 76M12.

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“…However, similarly to the results obtained in [17] for regular fluxes, the large-time behaviour of the entropy solution highly depends on the initial condition.…”

Section: Numerical Resultssupporting

confidence: 73%

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Martin

Vovelle²

2008

ESAIM: M2AN

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“…F (t, x, ., .) ρ,c π t,x (dρ)dxdt (33) holds along this subsequence, where <> ρ,c denotes expectation w.r.t. ν ρ,c .…”

Section: Average Entropy Inequalitymentioning

confidence: 99%

“…The hydrostatic limit follows from a uniqueness theorem that we establish for measure-valued stationary entropy solutions with boundary conditions. Such a result implies (and is actually equivalent to) asymptotic stability for entropy solutions with boundary conditions, a question studied so far only for convex ( [31,32]) or bell-shaped ( [33]) flux functions. We prove here such a result for general fluxes.…”

Section: Introductionmentioning

confidence: 98%

Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

Bahadoran

2012

Commun. Math. Phys.

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We consider attractive particle systems in Z d with product invariant measures. We prove that when particles are restricted to a subset of Z d , with birth and death dynamics at the boundaries, the hydrodynamic limit is given by the unique entropy solution of a conservation law, with boundary conditions in the sense of Bardos et al. ([7]). For the hydrostatic limit between parallel hyperplanes, we prove a multidimensional version of the phase diagram conjectured in [38], and show that it is robust with respect to perturbations of the boundaries.AMS 2000 subject classifications. 60K35, 82C22, 82C26; 35L65, 35L67, 35L50.

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Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

Cited by 2 publications

References 27 publications

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

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