Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates

Abstract

<jats:title>Abstract</jats:title> <jats:p>Our aim is to approximate a reference velocity field solving the two-dimensional Navier–Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniform-in-time error estimates between the approximation and the reference solution are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.</jats:p>

Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations

“…The recent paper [15] showed that CDA can be effectively used for weather prediction, showing that it can indeed be a powerful tool on practical large scale problems. Convergence of discretizations of CDA models was studied in [30,41,27,21] , and found results similar to those at the continuous level. Our interest in the CDA algorithm arises from its adaptability to a wide range of nonlinear problems, as well as its small computational cost and straight-forward implementation.…”

“…This seemingly minor change had profound impacts, and the authors of [5] were able to prove that using only sparse observations, the CDA algorithm applied to the 2D Navier-Stokes equations converges to the correct solution exponentially fast in time, independent of the choice initial data. This stimulated a large amount of recent research on the CDA algorithm; see, e.g., [3,6,7,10,11,13,17,18,19,20,21,22,27,31,30,35,40,41] and the references therein. The recent paper [15] showed that CDA can be effectively used for weather prediction, showing that it can indeed be a powerful tool on practical large scale problems.…”

Efficient dynamical downscaling of general circulation models using continuous data assimilation

“…Furthermore, stability analysis has enabled Ibdah et al . () and Mondaini and Titi () to establish uniform in time error estimates for the spatial discretization and the full discretization of the model, respectively, which makes its computational implementation reliable. Notably, it has also been observed that this CDA approach is equally applicable to other relevant dissipative systems, and that for certain systems it is sufficient to collect coarse‐mesh measurements of only part of the state variables (Farhat et al ., ; ; ; ).…”

Continuous data assimilation and long-time accuracy in a $C^0$ interior penalty method for the Cahn-Hilliard equation

“…This seemingly small change has led to a profound impact in accuracy and theory of data assimilation methods, as CDA provides mathematically rigorous justification of data assimilated solutions converging to true solutions exponentially fast in time (for arbitrarily inaccurate initial conditions), as well as long time accuracy and stability; these properties are unique among existing data assimilation techniques. CDA has so far been used to improve solutions in Navier-Stokes equations [35,8,21,34,33], with noisy data [7,24], and with temporal and spatial discretizations [30,39,27,25], for NS-α and Leray-α models [1,20], for Benard convection [2,19,22], for the Brinkman Forchheimer-extended Darcy [36] equation, for the surface quasi-geostrophic equation in [28], and for weather prediction [17], among others. Convergence of discretizations of CDA models has been studied in [30,39,44,27,25] for fluid related models, and it was found that if there is enough measurement data, then computed solutions will converge to the true solution exponentially fast in time, up to (optimal) discretization error.…”

“…CDA has so far been used to improve solutions in Navier-Stokes equations [35,8,21,34,33], with noisy data [7,24], and with temporal and spatial discretizations [30,39,27,25], for NS-α and Leray-α models [1,20], for Benard convection [2,19,22], for the Brinkman Forchheimer-extended Darcy [36] equation, for the surface quasi-geostrophic equation in [28], and for weather prediction [17], among others. Convergence of discretizations of CDA models has been studied in [30,39,44,27,25] for fluid related models, and it was found that if there is enough measurement data, then computed solutions will converge to the true solution exponentially fast in time, up to (optimal) discretization error.…”