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 nontrivial 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.
Similar content being viewed by others
References
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, Oxford University ess, 1961.
Curry, J. H.: Bounded Solutions of Finite Dimensional Approximations to the Boussinesq Equations, SIAM J. Math. Anal. 10 (1979), pp. 71–79.
Getling, A. V.: Rayleigh-B´enard Convection: Structures and Dynamics, Advanced Series in Nonlinear Dynamics 11, World Scientific, 1998.
Joseph, D. D.:On the Stability of the Boussinesq Equations, Arch. Rational Mech. Anal. 20 (1965), pp. 59–71.
Kearfott, R. B. and Kreinovich, V.: Applications of Interval Computations, Kluwer Academic Publishers, Dordrecht, 1996.
Nakao, M. T.:ANumericalVerificationMethod for the Existence ofWeak Solutions forNonlinear Boundary Value Problems, J. Math. Anal. Appl. 164 (1992), pp. 489–507.
Nakao, M. T.: Solving Nonlinear Elliptic Problems with Result Verification Using an H -1 Type Residual Iteration, Computing, Suppl. 9 (1993), pp. 161–173.
Rayleigh, J. W. S.: On Convection Currents in a Horizontal Layer of Fluid, When the Higher Temperature Is on the Under Side, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 6 32 (1916), pp. 529–546; and Scientific Papers 6 (1920), pp. 432–446.
Watanabe, Y., Yamamoto, N., Nakao, M. T., and Nishida, T.: A Numerical Verification of Nontrivial Solutions for theHeatConvection Problem, Journal of Mathematical FluidMechanics, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nakao, M.T., Watanabe, Y., Yamamoto, N. et al. Some Computer Assisted Proofs for Solutions of the Heat Convection Problems. Reliable Computing 9, 359–372 (2003). https://doi.org/10.1023/A:1025179130399
Issue Date:
DOI: https://doi.org/10.1023/A:1025179130399