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L∞ stability of the MUSCL methods

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Abstract

We present a general L ∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L ∞ stability.

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References

  1. Barth, T.J.: Numerical methods and error estimation for conservation laws on structured and unstructured meshes. In: VKI Computational Fluid Dynamics, Lecture Notes (2003)

  2. Barth, T.J., Ohlberger, M.: Finite volume methods: foundation and analysis. In: Encyclopedia of Computational Mechanics, vol. 1, chap. 15. Wiley, New York (2004)

  3. Buffard, T., Clain, S.: Monoslope and Multislope MUSCL Methods for Unstructured Meshes. J. Comput. Phys. doi:10.1016/j.jcp.2010.01.026 (2010)

  4. Chainais-Hillairet C.: Second order finite volume schemes for a nonlinear hyperbolic equation: error estimate. Math. Methods Appl. Sic. 23, 467–490 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Clain, S., Clauzon, V.: The multislope MUSCL method. In: Proceeding in the Finite Volumes for Complex Application, vol. 5, pp. 297–304. Wiley, New York (2008)

  6. Clauzon, V.: Analyse de Schémas d’ordre élevé pour les écoulements compressibles. Application à la simulation numérique d’une torche à plasma. PhD Thesis, Blaise Pascal University, Clermont Ferrand. http://tel.archives-ouvertes.fr/tel-00235951/fr/ (2008)

  7. Cockburn B., Hou S., Shu C.W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation law IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cournède, P.-H., Debiez, C., Dervieux, A.: A positive MUSCL scheme for triangulations. INRIA report 3465 (1998)

  9. Eymard R., Galloüet T., Herbin R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds) Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North Holland, Amsterdam (2000)

    Google Scholar 

  10. Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation law. In: Applied Mathematical Sciences, vol. 118. Springer, Berlin (1996)

  11. Goodman J.B., Leveque R.J.: On the accuracy of stable schemes for 2D scalar conversation laws. Math. Comput. 45(171), 15–21 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hubbard M.E.: Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155(1), 54–74 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jameson, A.: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows. AIAA paper no. 3359 (1993)

  14. Kröner D.: Numerical Schemes for Conservation Laws. Wiley Teubner, Chichester (1997)

    MATH  Google Scholar 

  15. Kröner D., Noelle S., Rokyta M.: Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71(4), 527–560 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Osher S.: Convergence of generalized MUSCL schemes. SIAM J. Numer. Anal. 22(5), 947–961 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Parka, J.S., Yoona, S.-H., Kim, C.: Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys. doi:10.1016/j.jcp.2009.10.011

  18. Spekreijse S.P.: Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws. Math. Comput. 49(179), 135–155 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  19. Van Leer B.: Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–376 (1974)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR CNRS 5219, 118 route de Narbonne, 31062, Toulouse cedex, France

    Stéphane Clain

  2. Laboratoire de Mathématiques, Université Clermont–Ferrand II, UMR CNRS 6620, 63177, Aubière cedex, France

    Vivien Clauzon

Authors
  1. Stéphane Clain
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  2. Vivien Clauzon
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Correspondence to Stéphane Clain.

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Clain, S., Clauzon, V. L∞ stability of the MUSCL methods. Numer. Math. 116, 31–64 (2010). https://doi.org/10.1007/s00211-010-0299-2

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  • DOI: https://doi.org/10.1007/s00211-010-0299-2

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