A numerical verification of nontrivial solutions for the heat convection problem

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

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalJournal of Mathematical Fluid Mechanics
Volume6
Issue number1
DOIs
Publication statusPublished - 2004
Externally publishedYes

Fingerprint

Computer-assisted Proof
Numerical Verification
Heat convection
Rounding error
Floating point
Steady-state Solution
Nontrivial Solution
Rayleigh
Convection
Fixed point theorem
convection
Heat
heat
Numerical Examples
Operator
newton
floating
theorems
operators

Keywords

  • Computer assisted proof
  • Fixed-point theorem
  • Rayleigh-Bénard convection

ASJC Scopus subject areas

  • Mathematical Physics
  • Condensed Matter Physics
  • Computational Mathematics
  • Applied Mathematics

Cite this

A numerical verification of nontrivial solutions for the heat convection problem. / Watanabe, Yoshitaka; Yamamoto, Nobito; Nakao, Mitsuhiro T.; Nishida, Takaaki.

In: Journal of Mathematical Fluid Mechanics, Vol. 6, No. 1, 2004, p. 1-20.

Research output: Contribution to journalArticle

Watanabe, Yoshitaka ; Yamamoto, Nobito ; Nakao, Mitsuhiro T. ; Nishida, Takaaki. / A numerical verification of nontrivial solutions for the heat convection problem. In: Journal of Mathematical Fluid Mechanics. 2004 ; Vol. 6, No. 1. pp. 1-20.
@article{a4edbba6ed4e41b580940765b021522c,
title = "A numerical verification of nontrivial solutions for the heat convection problem",
abstract = "A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-B{\'e}nard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.",
keywords = "Computer assisted proof, Fixed-point theorem, Rayleigh-B{\'e}nard convection",
author = "Yoshitaka Watanabe and Nobito Yamamoto and Nakao, {Mitsuhiro T.} and Takaaki Nishida",
year = "2004",
doi = "10.1007/s00021-003-0077-3",
language = "English",
volume = "6",
pages = "1--20",
journal = "Journal of Mathematical Fluid Mechanics",
issn = "1422-6928",
publisher = "Birkhauser Verlag Basel",
number = "1",

}

TY - JOUR

T1 - A numerical verification of nontrivial solutions for the heat convection problem

AU - Watanabe, Yoshitaka

AU - Yamamoto, Nobito

AU - Nakao, Mitsuhiro T.

AU - Nishida, Takaaki

PY - 2004

Y1 - 2004

N2 - A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

AB - A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

KW - Computer assisted proof

KW - Fixed-point theorem

KW - Rayleigh-Bénard convection

UR - http://www.scopus.com/inward/record.url?scp=10044237618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044237618&partnerID=8YFLogxK

U2 - 10.1007/s00021-003-0077-3

DO - 10.1007/s00021-003-0077-3

M3 - Article

VL - 6

SP - 1

EP - 20

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 1

ER -