Advertisement

Numerische Mathematik

, Volume 122, Issue 2, pp 227–278 | Cite as

Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes

  • Christophe BuetEmail author
  • Bruno Després
  • Emmanuel Franck
Article

Abstract

We propose an asymptotic preserving nodal discretization of the hyperbolic heat equation, also known as the P 1 equation, on unstructured meshes in 2-D. This method, in diffusive regime, overcomes the problem of the inconsistent limit with diffusion, of classical multidimensional extensions of 1-D asymptotic preserving schemes, based on edge formulation. We provide both theoretical and numerical results.

Mathematics Subject Classification (2010)

35L65 65M08 65M12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aavatsmark I., Eigestad G.: Numerical convergence of the MPFA O-method and U-method for general quadrilateral grids. Int. J. Numer. Meth. Fluids 51, 939–961 (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Allaire G.: Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, pp. 42. Oxford University Press, Oxford (2007)Google Scholar
  3. 3.
    Buet C., Cordier S., Lucquin-Desreux B., Mancini S.: Diffusion limit of the lorentz model: asymptotic preserving schemes. ESAIM:M2AN 32(4), 631–655 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buet C., Després B.: Asymptotic preserving and positive schemes for radiation hydrodynamics. JCP 215(2), 717–740 (2006)zbMATHGoogle Scholar
  5. 5.
    Breil J., Maire P.-H.: A cell-centered diffusion scheme on two-dimensional unstructured meshes. JCP 224, 785–823 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brunner, T.: Riemann solvers for time-dependent transport based on the maximum entropy and spherical harmonics closures. PhD thesis, Los AlamosGoogle Scholar
  7. 7.
    Carré G., Del Pino S., Desprès B., Labourasse E.: A Cell-centered lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. JCP 228(14), 5160–5183 (2009)zbMATHGoogle Scholar
  8. 8.
    Degond P., Deluzet F., Sangam A., Vignal M.-H.: An asymptotic preserving scheme for the Euler equations in a strong magnetic field. J. Comput. Phys. 228, 3540–3558 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Desprès B.: Weak consistency of the cell centered lagrangian GLACE scheme on general mesh in any dimension. Comput. Methods Appl. Mech. Eng 199(41–44), 2669–2679 (2010)zbMATHCrossRefGoogle Scholar
  10. 10.
    Droniou J., Eymard R., Gallouet T., Herbin R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. (M3AS) 20(2), 265–295 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Droniou J., Le-Potier C.: Construction and convergence study of local-maximum-principe preserving schemes for elliptic equations. SIAM J. Numer. Anal. 49, 459–490 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dubroca, B., Feugeas, J.L.: Hiérarchie des modèles aux moments pour le transfert radiatif. C. R. Acad. Sci. Paris t.329, Serie I, pp. 915–920 (1999)Google Scholar
  13. 13.
    Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problem on general nonconforming meshes. IMA J. Num. Anal. (2009)Google Scholar
  14. 14.
    Gosse L., Toscani G.: An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Acad. Sci. Paris Ser. I 334, 337–342 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Greenberg, J., Leroux, A.Y.: A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal 33(1), (1996)Google Scholar
  16. 16.
    Hirsch C.: Numerical computation of internal and external flows, vol. 1. Butterworth Heinemann, Oxford (2007)Google Scholar
  17. 17.
    Jin S., Levermore D.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. JCP 126, 449–467 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kershaw D.: Differencing of the diffusion equation in lagrangian hydrodynamic codes. JCP 39, 375–395 (1981)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kluth G., Després B.: Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. J. Comp. Phys. 229, 9092–9118 (2010)CrossRefGoogle Scholar
  20. 20.
    Lemou, M., Mieussens, L.: A new symptotic preserving scheme based on micro–macro formulation for linear kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 31(1), 334–368Google Scholar
  21. 21.
    Lipnikov K., Shashkov M., Svyatskiy D., Vassilevski Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. JCP 227, 492–512 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lowrie R.B., Morel J.E.: Methods for hyperbolic systems with stiff relaxation. Int. J. Num. Methods Fluids 40, 413–423 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mazeran, C.: Sur la structure mathématique et l’approximation numérique de l’hydrodynamique lagrangienne bidimensionnelle. PhD thesis, University of Bordeaux (2007)Google Scholar
  24. 24.
    Maire P-H., Abgrall R., Breil J., Ovadia J.: A cell-centered lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput. 29(4), 1781–1824 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Polyanin A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman Hall, Boca Raton (2002)zbMATHGoogle Scholar
  26. 26.
    Sheng, Z., Yue, J., Yuan, G.: Monotone finite volume schemes of non-equilibrium radiation diffusion equations of distorted meshes. SIAM J. Sci. Comput. 31(4), 2915–2934Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Christophe Buet
    • 1
    Email author
  • Bruno Després
    • 2
  • Emmanuel Franck
    • 1
    • 2
  1. 1.CEA, DAM, DIFArpajon CedexFrance
  2. 2.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations