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Journal of Nonlinear Science

, Volume 27, Issue 3, pp 1065–1087 | Cite as

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

  • Aseel Farhat
  • Evelyn Lunasin
  • Edriss S. Titi
Article

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.

Keywords

Bénard convection Boussinesq system Continuous data assimilation Signal synchronization Nudging Downscaling 

Mathematics Subject Classification

35Q30 93C20 37C50 76B75 34D06 

Notes

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Aseel Farhat
    • 1
  • Evelyn Lunasin
    • 2
  • Edriss S. Titi
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsUnited States Naval AcademyAnnapolisUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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