Some computer assisted proofs for solutions of the heat convection problems

Mitsuhiro T. Nakao, Yoshitaka Watanabe, Nobito Yamamoto, Takaaki Nishida

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

This is a continuation of our previous results (Y. Watanabe, N. Yamamoto, T. Nakao, and T. Nishida, "A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem," to appear in the Journal of Mathematical Fluid Mechanics). In that work, the authors considered two-dimensional Rayleigh-Bénard convection and proposed an approach to prove existence of steady-state solutions based on an infinite dimensional fixed-point theorem using a Newton-like operator with spectral approximation and constructive error estimates. We numerically verified several exact non-trivial solutions which correspond to solutions bifurcating from the trivial solution. This paper shows more detailed results of verification for given Prandtl and Rayleigh numbers. In particular, we found a new and interesting solution branch which was not obtained in the previous study, and it should enable us to present important information to clarify the global bifurcation structure. All numerical examples discussed are take into account of the effects of rounding errors in the floating point computations.

Original languageEnglish
Pages (from-to)359-372
Number of pages14
JournalReliable Computing
Volume9
Issue number5
DOIs
Publication statusPublished - 2003 Oct
Externally publishedYes

Fingerprint

Computer-assisted Proof
Heat convection
Nontrivial Solution
Convection
Heat
Spectral Approximation
Numerical Verification
Global Bifurcation
Fluid Mechanics
Rounding error
Rayleigh number
Prandtl number
Floating point
Steady-state Solution
Rayleigh
Continuation
Fixed point theorem
Error Estimates
Trivial
Branch

ASJC Scopus subject areas

  • Software
  • Computational Mathematics
  • Applied Mathematics

Cite this

Some computer assisted proofs for solutions of the heat convection problems. / Nakao, Mitsuhiro T.; Watanabe, Yoshitaka; Yamamoto, Nobito; Nishida, Takaaki.

In: Reliable Computing, Vol. 9, No. 5, 10.2003, p. 359-372.

Research output: Contribution to journalArticle

Nakao, MT, Watanabe, Y, Yamamoto, N & Nishida, T 2003, 'Some computer assisted proofs for solutions of the heat convection problems', Reliable Computing, vol. 9, no. 5, pp. 359-372. https://doi.org/10.1023/A:1025179130399
Nakao, Mitsuhiro T. ; Watanabe, Yoshitaka ; Yamamoto, Nobito ; Nishida, Takaaki. / Some computer assisted proofs for solutions of the heat convection problems. In: Reliable Computing. 2003 ; Vol. 9, No. 5. pp. 359-372.
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