We study numerical schemes for incompressible Navier-Stokes equations using IMEX temporal discretizations, finite element spacial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical. IntroductionData assimilation (DA) refers to a wide class of schemes for incorporating observational data in simulations, in order to increase the accuracy of solutions and to obtain better estimates of initial conditions. It is the subject of a large body of work (see, e.g., [13,31,33], and the references therein). DA algorithms are widely used in weather modeling, climate science, and hydrological and environmental forecasting [31]. Classically, these techniques are based on linear quadratic estimation, also known as the Kalman Filter. The Kalman Filter is described in detail in several textbooks, including [13,31,33,10], and the references therein.Recently, a promising new approach to data assimilation was pioneered by Azouani, Olson, and Titi [3,4] (see also [9,25,39] for early ideas in this direction). This new approach, which we call AOT Data Assimilation or continuous data assimilation, adds a feedback control term at the PDE level that nudges the computed solution towards the reference solution corresponding to the observed data. A similar approach is taken by Blömker, Law, Stuart, and Zygalakis in [7] in the context of stochastic differential equations. The AOT algorithm is based on feedback control at the PDE (partial differential equation) level, described below. The first works in this area assumed noise-free observations, but [5] adapted the method to the case of noisy data, and [19] adapted to the case in which measurements are obtained discretely in time and may be contaminated by systematic errors. Computational experiments on the AOT algorithm and its variants were carried * An important property of the b operator is that b(u, v, v) = 0 for u, v ∈ X.We will utilize the following bounds on b.Lemma 2.1. There exists a constant M > 0 dependent only on Ω satisfyingfor all u, v, w ∈ X for which the norms on the right hand sides are finite.Remark 2.2. Here and throughout, sharper estimates are possible if we restrict to 2D. However, for simplicity and generality, we do not make this restriction.Proof. These well known bounds follow from Hölder's inequality, Sobolev inequalities, and the Poincaré inequality. Discretization preliminariesDenote by τ h a regular, conforming triangulation of the domain Ω, and let X h ⊂ X, Q h ⊂ Q be an inf-sup stable pair of discrete velocity -pressure spaces. For simplicity, we will take X h = X ∩ P k and Q h = Q ∩...
We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.
We introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L 2 projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier-Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long-time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems. KEYWORDS continuous data assimilation, finite element method, Navier-Stokes equations 1 INTRODUCTION Data assimilation (DA) algorithms are widely used in weather prediction, climate modeling, and many other applications [35]. The term DA generally refers to schemes that incorporate observational data in simulations in order to increase accuracy and/or obtain better initial conditions. There is a large amount of literature on the general topic of DA [13, 35, 38], and several different techniques exist, such as the
We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in H 3 (Ω), under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as 2010 Mathematics Subject Classification. 35A01,
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