Numerical Viscosity and Convergence of Finite Volume Methods for Conservation Laws with Boundary Conditions

Benharbit, Saad; Chalabi, A.; Vila, Jean‐Paul

doi:10.1137/0732036

Cited by 28 publications

(25 citation statements)

References 17 publications

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“…To prove this result, one needs an estimate on the variation of the approximate solution, it uses an estimate on discrete H l norm of the approximate solution of the elliptic équation, this result in [5] is given by the finite element framework. Other results on the existence and the uniqueness of solutions of hyperbolic équa-tions are given in [7], [3], [1], [10].…”

Section: Introductionmentioning

confidence: 99%

Convergence of a finite volume scheme for an elliptic-hyperbolic system

Vignal¹

1996

ESAIM: M2AN

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Section: Introductionmentioning

confidence: 99%

Convergence of a finite volume scheme for an elliptic-hyperbolic system

Vignal¹

1996

ESAIM: M2AN

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“…DiPerna [11] showed that measure valued solutions are useful to prove convergence of approximations to scalar conservation laws: convergence follows by verifying that the approximations are uniformly bounded in L ∞ , weakly consistent with all entropy inequalities and consistent with the inititial data, cf. also [3], [4], [10], [15] and [19]. The work [18] extended DiPerna's result to include boundary conditions based on Bardos, LeRoux and Nedelec's boundary conditions for the Kruzkov entropies, derived in [2] to establish uniqueness and existence of solutions with bounded variation.…”

mentioning

confidence: 84%

“…and that ν and σ are Young measure solutions to (1)(2)(3), in the sense of Definition 2.2, then the contraction…”

mentioning

confidence: 99%

Scalar Conservation Laws with Boundary Conditions and Rough Data Measure Solutions

Moussa¹,

Szepessy²

2002

Methods and Applications of Analysis

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Abstract. Uniqueness and existence of L ∞ solutions to initial boundary value problems for scalar conservation laws, with continuous flux functions, is derived by L 1 contraction of Young measure solutions. The classical Kruzkov entropies, extended in Bardos, LeRoux and Nedelec's sense to boundary value problems, are sufficient for the contraction. The uniqueness proof uses the essence of Kruzkov's idea with his symmetric entropy and entropy flux functions, but the usual doubling of variables technique is replaced by the simpler fact that mollified measure solutions are in fact smooth solutions. The mollified measures turn out to have not only weak but also strong boundary entropy flux traces. Another advantage with the Young measure analysis is that the usual assumption of Lipschitz continuous flux functions can be relaxed to continuous fluxes, with little additional work.1. Background to Scalar Conservation Laws with Boundary Conditions. DiPerna [11] showed that measure valued solutions are useful to prove convergence of approximations to scalar conservation laws: convergence follows by verifying that the approximations are uniformly bounded in L ∞ , weakly consistent with all entropy inequalities and consistent with the inititial data, cf. also The present work, with continuous flux functions shows that the Kruzkov entropies, in Bardos, LeRoux and Nedelec's sense, are sufficient for L 1 contraction of Young measure solutions, which in turn implies uniqueness of L ∞ solutions. The uniqueness proof uses the essence of Kruzkov's idea with his symmetric entropy and entropy flux functions, but the usual doubling of variables technique is replaced by the simpler fact that mollified measure solutions are in fact smooth solutions. The mollified measures turn out to also have strong boundary entropy flux traces. Existence and uniqueness for the pure initial value problem, with continuous flux functions, was established by semi group methods in [9] and by measure solutions in [20].At the hart of the matter of initial boundary value problems to scalar conservation laws is the trace of entropy fluxes, which define the boundary condition. The first study [2] used solutions with bounded variation and hence their trace exist directly. The work [18] used the equation in the interior domain to show that the entropy flux, *

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“…It has also been applied to the numerical analysis of transport equations [6,7,9,11,19] since "Finite-Volume" schemes only give an L ∞ -estimate uniformly with respect to the mesh length of the numerical solution. Lastly, it has been extended to the Dirichlet problem for scalar conservation laws by introducing the notion of Young measure-trace [3,15,16,18].…”

Section: Entropy Process Solutionmentioning

confidence: 99%

Obstacle problems for scalar conservation laws

Levi¹

2001

ESAIM: M2AN

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Abstract. In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L ∞ . Lastly, we study the behaviour of this solution and its stability properties with respect to the associated obstacle functions. Résumé.

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Numerical Viscosity and Convergence of Finite Volume Methods for Conservation Laws with Boundary Conditions

Cited by 28 publications

References 17 publications

Convergence of a finite volume scheme for an elliptic-hyperbolic system

Convergence of a finite volume scheme for an elliptic-hyperbolic system

Scalar Conservation Laws with Boundary Conditions and Rough Data Measure Solutions

Obstacle problems for scalar conservation laws

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