Some computer assisted proofs for solutions of the heat convection problems

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

    Research output: Contribution to journalArticlepeer-review

    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 - Oct 2003

    All Science Journal Classification (ASJC) codes

    • Software
    • Computational Mathematics
    • Applied Mathematics

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