Journal of Scientific Computing

, Volume 43, Issue 3, pp 388–401

Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems

  • Mitsuhiro T. Nakao
  • Yoshitaka Watanabe
  • Nobito Yamamoto
  • Takaaki Nishida
  • Myoung-Nyoung Kim
Article

DOI: 10.1007/s10915-009-9303-3

Cite this article as:
Nakao, M.T., Watanabe, Y., Yamamoto, N. et al. J Sci Comput (2010) 43: 388. doi:10.1007/s10915-009-9303-3

Abstract

In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.

Keywords

Navier-Stokes equationNonlinear heat convectionBifurcation pointComputer assisted proof

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
  • Yoshitaka Watanabe
    • 2
  • Nobito Yamamoto
    • 3
  • Takaaki Nishida
    • 4
  • Myoung-Nyoung Kim
    • 1
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan
  3. 3.Department of Computer Science and Information MathematicsThe University of Electro-CommunicationsTokyoJapan
  4. 4.Faculty of Science and EngineeringWaseda UniversityTokyoJapan