A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems

Myoungnyoun Kim, Mitsuhiro T. Nakao, Yoshitaka Watanabe, Takaaki Nishida

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

This paper is the three dimensional extension of the two dimensional work in Nakao et al. (Reliable Comput 9(5):359-372, 2003) and Watanabe et al. (J Math Fluid Mech 6:1-20, 2004) on a computer assisted proof of the existence of nontrivial steady state solutions for Rayleigh-Bénard convection based on the fixed point theorem using a Newton like operator. The differences are emerging of complicated types of bifurcation, direct attack on the problem without stream functions, and increased complexity of numerical computation. The last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case. Actually, this work should be the first result for the three dimensional Navier-Stokes problems which seems to be very difficult to solve by theoretical approaches.

Original languageEnglish
Pages (from-to)389-406
Number of pages18
JournalNumerische Mathematik
Volume111
Issue number3
DOIs
Publication statusPublished - 2009 Jan
Externally publishedYes

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Numerical Verification
Rayleigh
Three-dimensional
Fluids
Computer-assisted Proof
Navier-Stokes Problem
Stream Function
Steady-state Solution
Bifurcation Diagram
Numerical Computation
Convection
Fixed point theorem
Bifurcation
Attack
Fluid
Operator

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. / Kim, Myoungnyoun; Nakao, Mitsuhiro T.; Watanabe, Yoshitaka; Nishida, Takaaki.

In: Numerische Mathematik, Vol. 111, No. 3, 01.2009, p. 389-406.

Research output: Contribution to journalArticle

Kim, Myoungnyoun ; Nakao, Mitsuhiro T. ; Watanabe, Yoshitaka ; Nishida, Takaaki. / A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. In: Numerische Mathematik. 2009 ; Vol. 111, No. 3. pp. 389-406.
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