Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albanez, D., Nussenzveig-Lopes, H., Titi, E.S.: Continuous data assimilation for the three-dimensional Navier–Stokes-\(\alpha \) model. Asymptot. Anal. 97(1–2), 139–164 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Altaf, M.U., Titi, E.S., Gebrael, T., Knio, O., Zhao, L., McCabe, M.F., Hoteit, I.: Downscaling the 2D Bénard convection equations using continuous data assimilation. Computational Geosciences (COMG) (2015). arXiv:1512.04671

  • Azouani, A., Titi, E.S.: Feedback control of nonlinear dissipative systems by finite determining parameters: a reaction-diffusion paradigm. Evol. Equ. Control Theory 3(4), 579–594 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Azouani, A., Olson, E., Titi, E.S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24(2), 277–304 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Bessaih, H., Olson, E., Titi, E.S.: Continuous assimilation of data with stochastic noise. Nonlinearity 28, 729–753 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Cao, C., Kevrekidis, I., Titi, E.S.: Numerical criterion for the stabilization of steady states of the Navier–Stokes equations. Indiana Univ. Math. J. 50, 37–96 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Charney, J., Halem, J., Jastrow, M.: Use of incomplete historical data to infer the present state of the atmosphere. J. Atmos. Sci. 26, 1160–1163 (1969)

    Article  Google Scholar 

  • Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1988)

  • Farhat, A., Jolly, M.S., Titi, E.S.: Continuous data assimilation for the 2D Bénard convection through velocity measurements alone. Phys. D 303, 59–66 (2015)

    Article  MathSciNet  Google Scholar 

  • Farhat, A., Lunasin, E., Titi, E.S.: Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field. J. Math. Fluid Mech. 18(1), 1–23 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Farhat, A., Lunasin, E., Titi, E.S.: A note on abridged continuous data assimilation for the 3D subgrid scale \(\alpha \)-models of turbulence, Preprint

  • Farhat, A., Lunasin, E., Titi, E.S.: Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements. J. Math. Anal. Appl. 438(1), 492–506 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Foias, C., Manley, O., Temam, R.: Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal. Theory Methods Appl. 11, 939–967 (1987)

    Article  MATH  Google Scholar 

  • Foias, C., Mondaini, C., Titi, E.S.: A discrete data assimilation scheme for the solutions of the 2D Navier–Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. 15(4), 2109–2142 (2000)

    Article  MATH  Google Scholar 

  • Gesho, M., Olson, E., Titi, E.S.: A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations. Commun. Comput. Phys. 19(4), 1094–1110 (2016)

    MathSciNet  Google Scholar 

  • Ghil, M., Shkoller, B., Yangarber, V.: A balanced diagnostic system compatible with a barotropic prognostic model. Mon. Weather Rev. 105, 1223–1238 (1977)

    Article  Google Scholar 

  • Ghil, M., Halem, M., Atlas, R.: Time-continuous assimilation of remote-sounding data and its effect on weather forecasting. Mon. Weather Rev. 107, 140–171 (1978)

    Article  Google Scholar 

  • Jones, D.A., Titi, E.S.: Determining finite volume elements for the 2D Navier–Stokes equations. Phys. D 60, 165–174 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Leunberger, D.: An introduction to observers. IEEE. Trans. Autom. Control 16, 596–602 (1971)

    Article  Google Scholar 

  • Lunasin, E., Titi, E.S.: Finite determining parameters feedback control for distributed nonlinear dissipative systems: a computational study, arXiv:1506.03709 [math.AP] (2015)

  • Markowich, P., Titi, E.S., Trabelsi, S.: Continuous data assimilation for the three-dimensional Brinkman–Forchheimer–Extended Darcy model. Nonlinearity 29(4), 1292–1328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Nijmeijer, H.: A dynamic control view of synchronization. Phys. D 154, 219–228 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, J.C.: Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)

  • Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, vol. 68. Springer, New York (1997)

    Book  Google Scholar 

  • Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, second ed., CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995)

  • Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI (2001), Reprint of the 1984 edition

  • Thau, F.E.: Observing the state of non-linear dynamic systems. Int. J. Control 17, 471–479 (1973)

    Article  MATH  Google Scholar 

Download references

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.

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evelyn Lunasin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edriss S. Titi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aseel Farhat.

Additional information

Communicated by Paul Newton.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Mathematics Subject Classification

Search

Navigation