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High–Order Asymptotic–Preserving Methods for Nonlinear Relaxation from Hyperbolic Systems to Convection–Diffusion Equations

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High Order Nonlinear Numerical Schemes for Evolutionary PDEs

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 99))

Abstract

In Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761) the authors propose an asymptotic–preserving method based on Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes which are adopted to deal with a class of nonlinear hyperbolic systems containing nonlinear diffusive relaxation. These schemes are able to solve such systems with no stiff nor parabolic restriction on the time step. In the limit when the relaxation parameter vanishes, the proposed scheme relaxes to a semi-implicit scheme for the limit nonlinear diffusion equation, thus overcoming the classical parabolic CFL condition in the time step. In this paper we consider an extension of the numerical treatment of the Kawashima-LeFloch’s model (LeFloch and Kawashima, private communication) proposed in Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761). We show that the same schemes introduced in Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761) relaxes to an semi-implicit scheme for the limit nonlinear convection-diffusion equation. A numerical example confirms the robustness and the accuracy of the scheme in order to capture the correct behavior of the solution in the hyperbolic–to–parabolic regime.

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Acknowledgements

This work has been partially supported by the Italian Project PRIN 2009 “Innovative numerical methods for hyperbolic problems with applications to fluid dynamics, kinetic theory and computational biology”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    Sebastiano Boscarino & Giovanni Russo

Authors
  1. Sebastiano Boscarino
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  2. Giovanni Russo
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Corresponding author

Correspondence to Giovanni Russo .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Zurich, Zurich, Switzerland

    Rémi Abgrall

  2. INRIA and Institut Polytechnique de Bordeaux, Institut de Mathématiques de Bordeaux, Talence, France

    Héloïse Beaugendre

  3. INRIA, Institut de Mathématiques de Bordeaux, Talence, France

    Pietro Marco Congedo

  4. INRIA and Institut Polytechnique de Bordeaux, Institut de Mathématiques de Bordeaux, Talence, France

    Cécile Dobrzynski

  5. INRIA, Laboratoire de Mathématiques et de leurs Applications, Pau, France

    Vincent Perrier

  6. INRIA, Institut de Mathématiques de Bordeaux, Talence, France

    Mario Ricchiuto

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Boscarino, S., Russo, G. (2014). High–Order Asymptotic–Preserving Methods for Nonlinear Relaxation from Hyperbolic Systems to Convection–Diffusion Equations. In: Abgrall, R., Beaugendre, H., Congedo, P., Dobrzynski, C., Perrier, V., Ricchiuto, M. (eds) High Order Nonlinear Numerical Schemes for Evolutionary PDEs. Lecture Notes in Computational Science and Engineering, vol 99. Springer, Cham. https://doi.org/10.1007/978-3-319-05455-1_1

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