Advertisement

The European Physical Journal Special Topics

, Volume 146, Issue 1, pp 235–247 | Cite as

Chebyshev collocation methods in thermoconvective problems

  • H. HerreroEmail author
Article
  • 63 Downloads

Abstract.

Chebyshev collocation methods are high-order methods. This means that high precision is obtained with low-order expansions. Then `small' matrices appear in the numerical implementation and reduced computing resources become necessary. Thermoconvective fluid dynamics problems are large ones, involving various partial differential equations for several fields in large dimensions. The models present a number of difficulties, such as the different orders of derivatives for the different fields, or lack of information on the boundary conditions for pressure. This paper presents a review of the specific characteristics of this method when it is applied to thermoconvective problems: the method, the types of problems, the different resulting models, the strategies to overcome the difficulties of the different derivative orders and the characteristics of the matrices, and the convergence properties. A comparison of its efficiency with other numerical methods like finite differences and finite elements is included.

Keywords

Rayleigh Number European Physical Journal Special Topic Linear Stability Analysis Marangoni Number Arnoldi Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Ahlers, Physica D 51, 421 (1991) Google Scholar
  2. H. Bénard, Rev. Gén. Sci. Pures Appl. 11, 1261 (1900) Google Scholar
  3. C. Bernardi, Y. Maday, Approximations Spectrales De Problémes Aux Limites Elliptiques (Springer-Verlag, Berlin, 1991) Google Scholar
  4. C. Bernardi, C. Canuto, Y. Maday, SIAM J. Numer. Anal. 25, 1237 (1988) Google Scholar
  5. C. Bernardi, Y. Maday, SIAM J. Numer. Anal. 26, 769 (1989) Google Scholar
  6. B. Birnir, N. Svanstedt, Disc. Cont. Dyn. Syst. 10, 53 (2004) Google Scholar
  7. E. Bodenschatz, J.R. de Bruyn, G. Ahlers, D.S. Cannell, Phys. Rev. Lett. 67, 3078 (1991) Google Scholar
  8. C. Canuto, A. Quarteroni, Spectral Methods for Partial Differential Equations, edited by R.G. Voigt, D. Gottlieb, M.Y. Hussaini (SIAM-CBMS, Philadelphia, 1984), pp. 55–78 Google Scholar
  9. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988) Google Scholar
  10. W.M. Cao, R.D. Haynesn, M.R. Trummer, J. Sci. Comp. 24, 343 (2005) Google Scholar
  11. J.Y. Chemin, Jean Jeray et Navier-Stokes. Numéro spécial de la Gazette des Mathématiciens, Societé Mathématique de France (2000), pp. 71–82 Google Scholar
  12. K.A. Cliffe, T.J. Garrat, A. Spence, SIAM J. Matrix Anal. Appl. 5, 1310 (1994) Google Scholar
  13. M.G. Crandall, P.H. Rabinowitz, J. Funct. Anal. 8, 321 (1971) Google Scholar
  14. P.C. Dauby, G. Lebon, J. Fluid Mech. 329, 25 (1996) Google Scholar
  15. F. Daviaud, J.M. Vince, Phys. Rev. E 48, 4432 (1993) Google Scholar
  16. M.O. Deville, Comp. Methods Appl. Mech. Eng. 80, 27 (1990) Google Scholar
  17. N. Garnier, A. Chiffaudel, Eur. Phys. J. B 19, 87 (2001) Google Scholar
  18. T.J. Garrat, G. Moore, A. Spence, Contibutions in Numerical Mathematics (World Sci. Publ., Singapore, 1993) Google Scholar
  19. V. Girault, P.A. Raviart, Finite Element Methods for Navier–Stokes Equations (Springer-Verlag, Berlin, 1986) Google Scholar
  20. G.H. Golub, C.F. van Loan, Matrix Computations (John Hopkins Univ. Press, Baltimore, 1989) Google Scholar
  21. M. González Burgos, Actas del XIV CEDYA/IV CMA Congress of Applied Mathematics (U. Barcelona, Vic, 1995) Google Scholar
  22. D. Gotlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, Philadelphia, 1993) Google Scholar
  23. P.M. Gresho, R.L. Sani, Int. J. Num. Meth. Fluids 7, 1111 (1987) Google Scholar
  24. H. Herrero, A.M. Mancho, Phys. Rev. E 57, 7336 (1998) Google Scholar
  25. H. Herrero, A.M. Mancho, Appl. Numer. Math. 33, 161 (2000) Google Scholar
  26. H. Herrero, S. Hoyas, A. Donoso, A.M. Mancho, J.M. Chacón, R.F. Portugués, B. Yeste, J. Scien. Comp. 8, 312 (2003) Google Scholar
  27. H. Herrero, A.M. Mancho, Int. J. Numer. Meth. Fluids 39, 391 (2002) Google Scholar
  28. H. Herrero, A.M. Mancho, S. Hoyas, Numerical Mathematics and Advanced Applications, Enumath 2005 (Springer, Berlin, 2006), pp. 889–896 Google Scholar
  29. S. Hoyas, H. Herrero, A.M. Mancho, J. Phys. A: Math. Gen. 35, 4067 (2002) Google Scholar
  30. S. Hoyas, H. Herreron, A.M. Mancho, Phys. Rev. E 66, 057301 (2002) Google Scholar
  31. S. Hoyas, H. Herreron, A.M. Mancho, CD Actas del XVII CEDYA/VII CMA (Salamanca, 2001) Google Scholar
  32. S. Hoyas, Estudio teŕico y numérico de un problema de convección de Bénard–Marangoni (Universidad Complutense de Madrid, 2003) Google Scholar
  33. S. Hoyas, A.M. Mancho, H. Herrero, N. Garnier, A. Chiffaudel. Phys. Fluids 17, 054104 (2005) Google Scholar
  34. K. Ito, S.S. Ravindran, SIAM J. Sci. Comput. 19, 1847 (1998) Google Scholar
  35. G. Kasperski, G. Lebrosse, Phys. Fluids 12, 2695 (2000) Google Scholar
  36. E.L. Koschmider, Bénard Cells and Taylor Vortices (Cambridge University Press, Cambridge, 1993) Google Scholar
  37. R.B. Lehoucq, D.C. Sorensen, SIAM J. Matrix Anal. Appl. 17, 789 (1996) Google Scholar
  38. J. Leray, Selected papers. Oeuvres scientifiques, Vol. II, Fluid dynamics and real partial differential equations. With an introduction by Peter Lax, edited by Paul Malliavin (Springer-Verlag, Berlin; Societé Mathématique de France, Paris, 1998) Google Scholar
  39. E. Leriche, G. Labrosse, SIAM J. Scien. Comp. 22, 1386 (2000) Google Scholar
  40. T. Ma, S. Wang, Comm. Pure Appl. Anal. 2, 591 (2003) Google Scholar
  41. Y. Maday, B. Métivet, Modél. Math. Anal. Numér. 21, 93 (1986) Google Scholar
  42. A. M. Mancho, H. Herrero, J. Burguete, Phys. Rev. E 56, 2916 (1997) Google Scholar
  43. A.M. Mancho, H. Herrero, Phys. Fluids 12, 1044 (2000) Google Scholar
  44. F. Marqués, M. Net, J.M. Massaquer, I. Mercader, Comp. Meth. Appl. Mech. Eng. 110, 157 (1993) Google Scholar
  45. I. Mercader, M. Net, A. Falques, Comp. Meth. Appl. Mech. Eng. 91, 1245 (1991) Google Scholar
  46. I. Mercader, J. Prat, E. Knobloch, Int. J. Bif. Chaos 11, 27 (2001) Google Scholar
  47. M.C. Navarro, H. Herrero, A.M. Mancho, Proc. Int. Conf. Math. and Stat. Modeling in Honor E. Castillo (U. Castilla-La Mancha, Ciudad Real, 2006) Google Scholar
  48. M.C. Navarro, A. Wathen, H. Herrero, A.M. Mancho, Comm. Comput. Phys. (submitted) Google Scholar
  49. M.C. Navarro, A.M. Mancho, H. Herrero, Chaos (submitted) Google Scholar
  50. M.C. Navarro, H. Herrero, S. Hoyas, Comp. Meth. Appl. Mech. Eng. (submitted) Google Scholar
  51. H.D. Nguyen, S. Paik, J.N. Chung, Num. Heat Transfer Part A: Applications 24, 161 (1993) Google Scholar
  52. D.A. Nield, J. Fluid Mech. 19, 341 (1964) Google Scholar
  53. J.R.A. Pearson, J. Fluid Mech. 4, 489 (1958) Google Scholar
  54. J.K. Platten, J.C. Legros, Convection in Liquids (Springer, Berlin, 1984) Google Scholar
  55. C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics (Oxford University Press, Oxford, 1997) Google Scholar
  56. P.H. Rabinowitz, Arch. Rational Mech. Anal. 29, 32 (1968) Google Scholar
  57. P. H. Rabinowitz, J. Functional Anal. 7, 487 (1971) Google Scholar
  58. Rayleigh (Lord), Phil. Mag. 32, 529 (1916) Google Scholar
  59. R.J. Riley, G.P. Neitzel, J. Fluid Mech. 359, 143 (1998) Google Scholar
  60. S. Rosenblat, G.M. Homsy, S.H. Davis, J. Fluid Mech. 120, 91 (1982) Google Scholar
  61. B.-C. Sim, A. Zebib, D. Schwabe, J. Fluid Mech. 491, 259 (2003) Google Scholar
  62. M.K. Smith, S.H. Davis, J. Fluid Mech. 132, 119 (1983) Google Scholar
  63. R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis (SIAM, Philadelphia, 1995) Google Scholar
  64. L.N. Trefethen, Spectral Methods in Matlab (SIAM, Philadelphia, 2000) Google Scholar
  65. S.H. Xin, P. Le Quere, Int. J. Num. Meth. Fluids 40, 981 (2002) Google Scholar
  66. S.H. Xin, P. Le Quere, L.S. Tuckerman, Phys. Fluids 10, 850 (1998) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Departamento de MatemáticasFacultad de Ciencias Químicas, Universidad de Castilla-La ManchaCiudad RealSpain

Personalised recommendations