Kinetic Decomposition of Approximate Solutions to Conservation Laws: Application to Relaxation and Diffusion-Dispersion Approximations*

Hwang, Seok Yoon; Tzavaras, Athanasios E.

doi:10.1081/pde-120004900

Cited by 44 publications

(38 citation statements)

References 19 publications

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“…To conclude this introduction, let us mention the recent work of Ben Moussa and Szepessy [6] in which the concept of measure-valued solution to deal with "very weak solutions" is used, and let us state some other occurrences of "kinetic methods" in the study of first-order problems with boundary conditions [5,20]; see also [21,13,14].…”

Section: Introduction Let ω Be a Strong Lipschitz Open Subset Of Rmentioning

confidence: 99%

A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

Imbert¹,

Vovelle²

2004

SIAM J. Math. Anal.

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Abstract. We state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution an entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub-and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (comparison theorem). We next study a Bhatnagar-Gross-Krook-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the comparison theorem.

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Section: Introduction Let ω Be a Strong Lipschitz Open Subset Of Rmentioning

confidence: 99%

A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

Imbert¹,

Vovelle²

2004

SIAM J. Math. Anal.

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“…LeFloch and Natalini [19] used the concept of measure-valued solution and established convergence results assuming that the diffusion dominates the dispersion. Subsequently, the approach of kinetic decomposition and velocity averaging [25] was introduced by Hwang and Tzavaras [11] to analyze singular limits including nonclassical shock waves. Furthermore, we also mention related works by Wu [32] and Jacobs, McKinney, and Shearer [12] which provides the first existence result of undercompressive shocks for the modified Korteweg-de Vries-Burgers equation.…”

Section: Definition 12 (Entropy Solutionmentioning

confidence: 99%

Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

2016

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Abstract. We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kružkov-Oleȋnik entropy solutions, but contain nonclassical undercompressive shock waves.

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“…Contractive kinetic equations provide a general framework for the extension of Kruzhkov theory to kinetic models. However, not every kinetic model with one conservation law generates an L 1 -contraction; see [11] for convergence results outside the Kruzhkov setting.…”

Section: Hydrodynamic and Diffusive Limits 531mentioning

confidence: 99%

Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves

Portilheiro

Tzavaras

2006

Trans. Amer. Math. Soc.

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Abstract. We consider a class of kinetic equations, equipped with a single conservation law, which generate L 1 -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the "dissipative" sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in L 1 . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.

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Kinetic Decomposition of Approximate Solutions to Conservation Laws: Application to Relaxation and Diffusion-Dispersion Approximations*

Cited by 44 publications

References 19 publications

A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves

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