Development of DDFV Methods for the Euler Equations

  • Christophe BerthonEmail author
  • Yves Coudière
  • Vivien Desveaux
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 4)


We propose to extend some recent gradient reconstruction, the so–called DDFV approaches, to derive accurate finite volume schemes to approximate the weak solutions of the 2D Euler equations. A particular attention is paid on the limitation procedure to enforce the required robustness property. Some numerical experiments are performed to highlight the relevance of the suggested MUSCL–DDFV technique.


Finite volume methods for hyperbolic problems Euler equations DDFV reconstruction MUSCL reconstruction Robustness 


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  1. 1.
    Andreianov, B., Boyer, F., Hubert, F.: Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numerical Methods for Partial Differential Equations 23(1), 145–195 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Berthon, C.: Stability of the MUSCL schemes for the Euler equations. Comm. Math. Sci 3, 133–158 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bouchut, F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004)zbMATHGoogle Scholar
  4. 4.
    Buffard, T., Clain, S.: Monoslope and multislope MUSCL methods for unstructured meshes. Journal of Computational Physics 229(10), 3745–3776 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Coudière, Y., Manzini, G.: The Discrete Duality Finite Volume Method for Convection-diffusion Problems. SIAM Journal on Numerical Analysis 47(6), 4163–4192 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Domelevo, K., Omnes, P.: A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Mathematical Modelling and Numerical Analysis 39(6), 1203–1249 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Godlewski, E., Raviart, P.A.: Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118. Springer-Verlag, New York (1996)Google Scholar
  8. 8.
    Harten, A., Lax, P., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM review pp. 35–61 (1983)Google Scholar
  9. 9.
    Hermeline, F.: A finite volume method for the approximation of diffusion operators on distorted meshes. Journal of computational Physics 160(2), 481–499 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hermeline, F.: Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes. Computer Methods in Applied Mechanics and Engineering 196(21-24), 2497–2526 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numerical Methods for Partial Differential Equations 18(5), 584–608 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge Univ Pr (2002)Google Scholar
  13. 13.
    Toro, E.: Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Verlag (2009)Google Scholar
  14. 14.
    Toro, E., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock waves 4(1), 25–34 (1994)zbMATHCrossRefGoogle Scholar
  15. 15.
    Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics 32(1), 101–136 (1979)Google Scholar
  16. 16.
    Van Leer, B.: A historical oversight: Vladimir P. Kolgan and his high-resolution scheme. Journal of Computational Physics (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christophe Berthon
    • 1
    Email author
  • Yves Coudière
    • 1
  • Vivien Desveaux
    • 1
  1. 1.Laboratoire de Mathématiques Jean LerayUMR 6629Nantes Cedex 3France

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