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Reduced Basis Method Applied to a Convective Instability Problem

  • Henar Herrero
  • Yvon Maday
  • Francisco Pla
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

Numerical reduced basis methods are instrumental to solve parameter dependent partial differential equations problems in case of many queries. Bifurcation and instability problems have these characteristics as different solutions emerge by varying a bifurcation parameter. Rayleigh-Bénard convection is an instability problem with multiple steady solutions and bifurcations by varying the Rayleigh number. In this paper the eigenvalue problem of the corresponding linear stability analysis has been solved with this method. The resulting matrices are small, the eigenvalues are easily calculated and the bifurcation points are correctly captured. Nine branches of stable and unstable solutions are obtained with this method in an interval of values of the Rayleigh number. Different basis sets are considered in each branch. The reduced basis method permits one to obtain the bifurcation diagrams with much lower computational cost.

Notes

Acknowledgements

This work was partially supported by the Research Grants MINECO (Spanish Government) MTM2015-68818-R, which include RDEF funds.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dpto. MatemáticasU. Castilla-La ManchaCiudad RealSpain
  2. 2.Laboratoire Jacques-Louis LionsSorbone Universités, UPMC Univ Paris 06, UMR 7598ParisFrance
  3. 3.Institut Universitaire de FranceParisFrance
  4. 4.Division of Applied MathsBrown UniversityProvidenceUSA

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