A Discrete Data Assimilation Scheme for the Solutions of the Two-Dimensional Navier-Stokes Equations and Their Statistics

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

[1]  R. Temam,et al.  Asymptotic analysis of the navier-stokes equations , 1983 .

[2]  Edriss S. Titi,et al.  Continuous data assimilation for the 2D Bénard convection through velocity measurements alone , 2014, 1410.1767.

[3]  Ciprian Foias,et al.  Estimates on enstrophy, palinstrophy, and invariant measures for 2-D turbulence ✩ , 2010 .

[4]  Ning Liu,et al.  Inverse Theory for Petroleum Reservoir Characterization and History Matching , 2008 .

[5]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[6]  Peter Korn,et al.  Data assimilation for the Navier–Stokes-α equations , 2009 .

[7]  Edriss S. Titi,et al.  Data Assimilation algorithm for 3D B\'enard convection in porous media employing only temperature measurements , 2015, 1506.08678.

[8]  Hakima Bessaih,et al.  Continuous data assimilation with stochastically noisy data , 2014, 1406.1533.

[9]  J. Charney,et al.  Use of Incomplete Historical Data to Infer the Present State of the Atmosphere , 1969 .

[10]  Edriss S. Titi,et al.  Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations , 2010, 1010.6105.

[11]  A. Bennett,et al.  Inverse Modeling of the Ocean and Atmosphere , 2002 .

[12]  R. Temam,et al.  Determination of the solutions of the Navier-Stokes equations by a set of nodal values , 1984 .

[13]  Edriss S. Titi,et al.  Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model , 2015, 1502.00964.

[14]  D. Luenberger.  An introduction to observers , 1971 .

[15]  Eugenia Kalnay,et al.  Atmospheric Modeling, Data Assimilation and Predictability , 2002 .

[16]  F. Thau.  Observing the state of non-linear dynamic systems† , 1973 .

[17]  Edriss S. Titi,et al.  Determining Modes for Continuous Data Assimilation in 2D Turbulence , 2003 .