Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions

Joseph, Kayyunnapara T.; LeFloch, Philippe G.

doi:10.1016/j.crma.2006.11.015

Numerical Analysis/Partial Differential Equations

Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions

Presented by: Olivier Pironneau

Joseph, Kayyunnapara T.¹; LeFloch, Philippe G.²

¹School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
²Laboratoire Jacques-Louis Lions & Centre national de la recherche scientifique, UMR 7598, Université Pierre et Marie Curie (Paris VI), 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 59-64

Abstract (Original language)
Résumé

We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system $\partial_{t} u_{ε} + A (u_{ε}) \partial_{x} u_{ε} = ε t \partial_{x} (B (u_{ε}) \partial_{x} u_{ε})$ , with $ε > 0$ . We assume that the matrix $A (u)$ is strictly hyperbolic and that the diffusion matrix satisfies $| B (u) - Id | ≪ 1$ . No genuine non-linearity assumption is required. We show the existence of a smooth, self-similar solution $u_{ε} = u_{ε} (x / t)$ which has bounded total variation, uniformly in the diffusion parameter $ε > 0$ . In the limit $ε \to 0$ , the functions $u_{ε}$ converge towards a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation $\partial_{t} u^{ε} + \partial_{x} v^{ε} = 0$ , $\partial_{t} v^{ε} + a^{2} B (u) \partial_{x} u^{ε} = (f (u^{ε}) - v^{ε}) / (ε t)$ . We also cover the boundary-value problem in a half-space for the same regularizations.

Nous considérons les approximations auto-semblables d'un système hyperbolique non-linéaire à une dimension d'espace avec donnée initiale de type « problème de Riemann », en particulier le système $\partial_{t} u_{ε} + A (u_{ε}) \partial_{x} u_{ε} = ε t \partial_{x} (B (u_{ε}) \partial_{x} u_{ε})$ , avec $ε > 0$ . Nous supposons que la matrice $A (u)$ est strictement hyperbolique et que la matrice de diffusion satisfait $| B (u) - Id | ≪ 1$ . Aucune hypothèse de « vraie non-linéarité » n'est imposée. Nous démontrons que ce problème admet une solution régulière, auto-semblable $u_{ε} = u_{ε} (x / t)$ de variation totale uniformément bornée par rapport au paramètre de diffusion $ε > 0$ . Lorsque $ε \to 0$ , les fonctions $u_{ε}$ convergent vers une solution du problème de Riemann associé au système hyperbolique. Nous établissons aussi un résultat analogue pour les approximations par relaxation données par $\partial_{t} u^{ε} + \partial_{x} v^{ε} = 0$ , $\partial_{t} v^{ε} + a^{2} B (u) \partial_{x} u^{ε} = (f (u^{ε}) - v^{ε}) / (ε t)$ . Ces résultats sont finalement étendus au problème de Riemann associé à ces mêmes régularisations et posé dans un demi-espace avec condition au bord.

Received: 2006-02-15
Accepted: 2006-11-13
Published online: 2006-12-15

DOI: 10.1016/j.crma.2006.11.015

Author's affiliations:

Joseph, Kayyunnapara T. ¹ ; LeFloch, Philippe G. ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
² Laboratoire Jacques-Louis Lions & Centre national de la recherche scientifique, UMR 7598, Université Pierre et Marie Curie (Paris VI), 75252 Paris cedex 05, France

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Joseph, Kayyunnapara T.; LeFloch, Philippe G. Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 59-64. doi: 10.1016/j.crma.2006.11.015

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