Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone
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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.
KeywordsBénard convection Boussinesq system Continuous data assimilation Signal synchronization Nudging Downscaling
Mathematics Subject Classification35Q30 93C20 37C50 76B75 34D06
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.
- 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
- Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1988)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, PreprintGoogle 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)
- 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)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)Google Scholar
- Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI (2001), Reprint of the 1984 editionGoogle Scholar