Convexity in Hyperbolic Problems. Application to a Discontinuous Galerkin Method for the Resolution of the Polydimensional Euler Equations

Bourdel, F.; Delorme, P.; Mazet, Pierre-Alain

doi:10.1007/978-3-322-87869-4_4

Cited by 9 publications

(5 citation statements)

References 2 publications

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“…In that case, the entropy inequality has a nonconservative form and is derived by using the notion of nonconservative product in Dal Maso-LeFloch-Murat, [23], and LeFloch-Liu, [41]. We first introduce some Theorem 3.2 provides a weak form of the entropy inequality (3.7).…”

Section: )mentioning

confidence: 99%

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Entropy flux‐splittings for hyperbolic conservation laws part I: General framework

LeFloch

1995

Comm Pure Appl Math

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A general framework is proposed for the derivation and analysis of flux-splittings and the corresponding flux-splitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing flux-splittings and to a method for the construction of entropy flux-splittings for general situations. A large family of genuine entropy flux-splittings is derived for several significant examples: the scalar conservation laws, the p-system, and the Euler system of isentropic gas dynamics. In particular, for the isentropic Euler system, we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy. For this system, it is proved that there exists a unique genuine entropy flux-splitting that satisfies all of the entropy inequalities, which is also the unique diagonatizable splitting. This splitting can be also derived by the so-called kinetic formulation. Simple and useful difference schemes are derived from the flux-splittings for hyperbolic systems. Such entropy flux-splitting schemes are shown to satisfy a discrete cell entropy inequality. For the diagonalizable splitting schemes, an a priori L" estimate is provided by applying the principle of bounded invariant regions. The convergence of entropy fluxsplitting schemes is proved for the 2 x 2 systems of conservation laws and the isentropic Euler system.

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Section: )mentioning

confidence: 99%

“…We now extend the result concerning the entropy consistency in Theorem 3.1 to arbitrary flux-splitting schemes. In that case, the entropy inequality has a nonconservative form and is derived by using the notion of nonconservative product in Dal Maso-LeFloch-Murat, [23], and LeFloch-Liu, [41]. We first introduce some notations.…”

Section: Indeed the Conditionsmentioning

confidence: 99%

Entropy flux‐splittings for hyperbolic conservation laws part I: General framework

LeFloch

1995

Comm Pure Appl Math

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“…[5] pour l'ensemble du contexte mathématique et numérique), avec les flux de Godunov [4], Osher [8] et Roe [10] et d'autre part les décompositions de flux ou schémas de « flux vector splitting ». Une décomposition de flux, avec Sanders-Prendergast [11], Van Leer [12], Bourdel et al [1] ou Perthame [9], suppose qu'on a pu écrire le flux R 3 W → f (W ) ∈ R 3 explicité en (1) sous la forme :…”

Section: Introductionunclassified

“…1 2 ± µ(ρ, p), p, ±ε(ρ, p) . De plus, pour une discontinuité de contact stationnaire, la viscosité numériqueV (W g , W d ) a pour expression V (W g , W d ) = µ(ρ d , p) − µ(ρ g , p), 0, ε(ρ d , p) − ε(ρ g , p).…”

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Décomposition de flux et discontinuité de contact stationnaire

Dubois

2000

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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“…(1.5). These inequalities were obtained independently by Bourdel-Delorme-Mazet [8] based on an analysis of the characteristics of the system (1.1), and by Benabdallah [5] for a specific system. The first result of existence for the initial-boundary value problem for a system was given by Benabdallah-Serre [6,7]: the vanishing viscosity method applied to the p-system of gas dynamics converges to a solution to (1.1) satisfying the set of inequalities (1.7).…”

Section: Introductionmentioning

confidence: 99%

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

Joseph

LeFloch

1999

Archive for Rational Mechanics and Analysis

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Abstract. This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L ∞ bounds on the approximate solutions and so dealing with L ∞ solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in "Boundary conditions for nonlinear hyperbolic systems of conservation laws", J. Diff. Equa. 71 (1988), 93-122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.

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Convexity in Hyperbolic Problems. Application to a Discontinuous Galerkin Method for the Resolution of the Polydimensional Euler Equations

Cited by 9 publications

References 2 publications

Entropy flux‐splittings for hyperbolic conservation laws part I: General framework

Entropy flux‐splittings for hyperbolic conservation laws part I: General framework

Décomposition de flux et discontinuité de contact stationnaire

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

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