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Reduced Basis Method Applied to a Convective Instability Problem

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Progress in Industrial Mathematics at ECMI 2016 (ECMI 2016)

Part of the book series: Mathematics in Industry ((TECMI,volume 26))

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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.

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Acknowledgements

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

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Authors and Affiliations

  1. Dpto. Matemáticas, U. Castilla-La Mancha, 13071, Ciudad Real, Spain

    Henar Herrero & Francisco Pla

  2. Laboratoire Jacques-Louis Lions, Sorbone Universités, UPMC Univ Paris 06, UMR 7598, F-75005, Paris, France

    Yvon Maday

  3. Institut Universitaire de France, Paris, France

    Yvon Maday

  4. Division of Applied Maths, Brown University, Providence, RI, USA

    Yvon Maday

Authors
  1. Henar Herrero
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  2. Yvon Maday
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  3. Francisco Pla
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Corresponding author

Correspondence to Francisco Pla .

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Editors and Affiliations

  1. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Peregrina Quintela

  2. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Patricia Barral

  3. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Dolores Gómez

  4. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Francisco J. Pena

  5. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Jerónimo Rodríguez

  6. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Pilar Salgado

  7. Department of Applied Mathematics, University of Santiago de Compostela, Lugo, Spain, Spain

    Miguel E. Vázquez-Méndez

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Herrero, H., Maday, Y., Pla, F. (2017). Reduced Basis Method Applied to a Convective Instability Problem. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_96

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