Roe solver with entropy corrector for uncertain hyperbolic systems

  • Authors:

  • J. Tryoen;O. Le Maítre;M. Ndjinga;A. Ern

  • Affiliations:

  • Université Paris Est, CERMICS, Ecole des Ponts, 77455 Marne la Vallée cedex 2, France and LIMSI-CNRS, 91403 Orsay cedex, France;LIMSI-CNRS, 91403 Orsay cedex, France;CEA-Saclay, DEN, DM2S, SFME, F-91191 Gif-sur-Yvette, France;Université Paris Est, CERMICS, Ecole des Ponts, 77455 Marne la Vallée cedex 2, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

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Abstract

This paper deals with intrusive Galerkin projection methods with a Roe-type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level. The aim of this paper is to design a cost-effective adaptation of the deterministic Dubois and Mehlman corrector to avoid entropy-violating shocks in the presence of sonic points. The adaptation relies on an estimate of the eigenvalues and eigenvectors of the Galerkin Jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface. We derive some indicators that can be used to decide where a correction is needed, thereby reducing the computational costs considerably. The effectiveness of the proposed corrector is assessed on the Burgers and Euler equations including sonic points.