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Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

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Abstract

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress-free boundary conditions on the walls, and the fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data are error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity.

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Acknowledgements

The work of A.F. is supported in part by the NSF Grant DMS-1418911. The work of E.L. is supported in part by the ONR Grant N0001416WX01475 and the ONR Grant N0001416WX00796. The work of E.S.T. is supported in part by the ONR Grant N00014-15-1-2333 and the NSF Grants DMS-1109640 and DMS-1109645.

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Aseel Farhat

  2. Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, USA

    Evelyn Lunasin

  3. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843-3368, USA

    Edriss S. Titi

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

Authors
  1. Aseel Farhat
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  2. Evelyn Lunasin
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  3. Edriss S. Titi
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Correspondence to Aseel Farhat.

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Communicated by Paul Newton.

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Farhat, A., Lunasin, E. & Titi, E.S. Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone. J Nonlinear Sci 27, 1065–1087 (2017). https://doi.org/10.1007/s00332-017-9360-y

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  • DOI: https://doi.org/10.1007/s00332-017-9360-y

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