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

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 condition on the walls, and 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 is 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. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

arXiv admin note: text overlap with arXiv:1506.08678

References in corpus (5)

• Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone
• Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model
• Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field
• Continuous Data Assimilation with Stochastically Noisy Data
• A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations

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• Data Assimilation in Large-Prandtl Rayleigh-Bénard Convection from Thermal Measurements
• An Efficient Continuous Data Assimilation Algorithm for the Sabra Shell Model of Turbulence