Transient<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>error estimates for well-balanced schemes on non-resonant scalar balance laws

Amadori, Debora; Gosse, Laurent

doi:10.1016/j.jde.2013.04.016

Cited by 12 publications

(5 citation statements)

References 38 publications

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“…For p ∼ p e , the WE method could be better effective since such a very strong decay does not appear to be available. A comparison of error estimates for the two methods, in the case of a scalar equation with source term, has been recently studied in [5].…”

Section: Introductionmentioning

confidence: 99%

Solutions for a hyperbolic model of multi-phase flow

Amadori

Corli

2013

ESAIM: Proc.

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We discuss a model for the flow of an inviscid fluid admitting liquid and vapor phases, as well as a mixture of them. The flow is modeled in one spatial dimension; the state variables are the specific volume, the velocity and the mass density fraction λ of vapor in the fluid. The equation governing the time evolution of λ contains a source term, which enables metastable states and drives the fluid towards stable pure phases.We first discuss, for the homogeneous system, the BV stability of Riemann solutions generated by large initial data and check the validity of several sufficient conditions that are known in the literature.Then, we review some recent results about the existence of solutions, which are globally defined in time, for λ close either to 0 or to 1 (corresponding to almost pure phases). These solutions possibly contain large shocks. Finally, in the relaxation limit, solutions are proved to satisfy a reduced system and the related entropy condition.Résumé. On discute un modèle pour l'écoulement d'un fluide non visqueux admettant phases liquides et de vapeur, ainsi qu'un mélange d'entre eux. L'écoulement est modélisé dans une dimension spatiale; les variables d'état sont le volume spécifique, la vitesse et la fraction de densité de masse λ de la vapeur dans le liquide. L'équation régissant l'évolution temporelle de λ contient un terme de source, ce qui permet desétats métastables et conduit le fluide vers de phases stables pures.Nous discutons d'abord, pour le système homogène, la stabilité BV des solutions de Riemann générés par des grandes données initiales et vérifions la validité de plusieurs conditions suffisantes qui sont connues dans la littérature.Ensuite, nous passons en revue quelques résultats récents sur l'existence de solutions, qui sont definies pour tous les temps, pour λ soit près de 0 ou de 1 (correspondantà des phases presque pures). Ces solutions sont susceptibles de contenir des grands chocs. Enfin, dans la limite de la relaxation, les solutions sont prouvèes satisfaire un système réduit et la condition d'entropie.

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Section: Introductionmentioning

confidence: 99%

Solutions for a hyperbolic model of multi-phase flow

Amadori

Corli

2013

ESAIM: Proc.

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“…The hydrostatic reconstruction scheme and its variants [4,24,18,17,16,15] is often used, and it is the subject of the present paper. Some results concerning consistency, stability and convergence of those schemes have been obtained in [9,10,13,26,8,1,2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

Bouchut

Lhébrard

2020

Math. Comp.

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We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

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“…The advantages of these techniques are numerous : by approximating accurately stationary solutions, it enables to handle stiff source terms. Note also that it improves the general convergence of the scheme, see [2][3][4][5] for precise estimates on that point.…”

Section: Numerical Methods For Chemotaxis Modelsmentioning

confidence: 99%

“…Although the properties of the scheme have been shown under the subcharacteristic condition, a numerical study of the previous scheme is also performed in [51] in the super-characteristic regime. Note that in [2][3][4][5], some precise and very useful error estimates on well-balanced schemes have been established for scalar balance laws and relaxation systems.…”

Section: Propertiesmentioning

confidence: 99%

“…• density and momentum of stationary solution (25) stay at rest, • starting from adapted initial data, the scheme is well-balanced for γ = 2 : W i+1 n = W i n for all i and for all n, • assuming that λ R = −λ L = λ, and under the CFL condition ∆t ε∆x max i∈Z |λ i+1/2 | ≤ 1 2 , the scheme is asymptotic preserving in the parabolic scaling with equation (2). Note that numerical results obtained with this scheme in [12] are similar to the ones presented in Subsection 2.2.…”

Section: Propertiesmentioning

confidence: 99%

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A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Ribot

2018

ESAIM: ProcS

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Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.

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TransientL1error estimates for well-balanced schemes on non-resonant scalar balance laws

Cited by 12 publications

References 38 publications

Solutions for a hyperbolic model of multi-phase flow

Solutions for a hyperbolic model of multi-phase flow

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

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