TY - JOUR

T1 - Some computer assisted proofs for solutions of the heat convection problems

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

AU - Yamamoto, Nobito

AU - Nishida, Takaaki

PY - 2003/10

Y1 - 2003/10

N2 - 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.

AB - 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.

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U2 - 10.1023/A:1025179130399

DO - 10.1023/A:1025179130399

M3 - Article

AN - SCOPUS:0041426696

VL - 9

SP - 359

EP - 372

JO - Reliable Computing

JF - Reliable Computing

SN - 1385-3139

IS - 5

ER -