Continuous data assimilation for the 2D Bénard convection through velocity measurements alone
Introduction
Accurate numerical simulations of nonlinear systems require high precision in the initial data. For most applications however, initial data which should ideally be defined on the whole physical domain, can be measured only discretely, often with inadequate resolution. Data assimilation refers to the process of completing, or enhancing the resolution of the initial condition. The classical method of continuous data assimilation, see, e.g., [1], is to insert observational measurements directly into a model as the latter is being integrated in time. The natural mathematical target for data assimilation is the global attractor. This set contains all the long time behavior; it is compact, invariant, and finite-dimensional. Another key notion is that of determining parameters. A projection (onto say a finite number of low Fourier modes, or other types of interpolant projections based on nodal values and volume elements) is said to be determining if, whenever the projection of two trajectories on the global attractor approach each other, as , the full trajectories approach each other, see, for example, [2], [3], [4], [5], [6], [7], [8], [9], [10] and references therein. One way to exploit this is to insert low mode observables from a time series into the equation for the evolution of the high modes. After a relatively short time interval the solution to the equation for the high modes is close to the high modes of the exact solution associated with the observables. At that point the low modes and high modes can be combined to form a complete good approximation of the state of the system at time , which can then be used as an initial condition for a high resolution simulation. This was the approach taken for the 2D Navier–Stokes in [11], [12], [13], [14], [15], [16], [17]. Except for the work in [12] for the 3DVAR Gaussian filter, and [18] using the determining parameters nudging approach of this paper for data assimilation, the previously mentioned theoretical work assumed that the observational measurements are error free. Notably, the authors of [14] present an algorithm for data assimilation that uses discrete in space and time measurements.
An alternative approach in [19] uses the observables in a feedback control term. The advantage is that, since no derivatives are required of the coarse grain observable, this works for a general class of interpolant operators. The main idea can be outlined in terms of a general evolutionary equation where the initial data is missing. Let represent an interpolant operator based on the spatial observations of the reference solution of system (1.1) at a coarse spatial resolution of size . Consider where is a relaxation (nudging) parameter, and is arbitrary. It is shown in [20] that if one takes large enough, and small enough (depending on ), then converges to the reference solution of the two-dimensional Navier–Stokes equations, at an exponential rate, as . An extension to this approach of [20] to the case where the observations are contaminated with random errors is studied in [18]. The feedback control approach to data assimilation plays a key role in the derivation in [21] of a determining form for the 2D NSE, which is an ordinary differential equation whose steady states are precisely the trajectories in the global attractor.
The Bénard convection problem is a model of the convection of an incompressible fluid layer in a box which is heated from below in such a way that the lower plate is maintained at a temperature while the upper one is at temperature , where and are constants. After a change of variables (see [22]), the non-dimensional two-dimensional Boussinesq equations that govern the velocity of the fluid, the pressure , and the normalized temperature (or the density) of the fluid read with the boundary conditions in the -direction and in the -direction, for simplicity, we will impose a periodicity condition
The Boussinesq system (1.3) is usually referred to as the Bénard convection problem. The global regularity of the two-dimensional Boussinesq equations was established in [23] (see also [24]) following the classical methods for the Navier–Stokes equations. The mathematical analysis of system (1.3) has been studied in [22] (see also [24]), where the existence and uniqueness of weak solutions in dimension two and three were proved, along with the existence of a finite dimensional global attractor in space dimension two. It was also shown in [22] that system (1.3) can be handled with different boundary conditions. For recent results concerning the two-dimensional Boussinesq equations, we refer to [25], [26], [27], [28], [29], [30], and references therein.
In this work, we present a new continuous data assimilation algorithm for the Bénard convection problem (1.3). The twist here is that we can recover the reference solution to (1.3) using coarse-grain data for the velocity alone; temperature data is not needed. This is done by solving with the boundary conditions
Here, is a modified pressure, and as for (1.2), may be chosen arbitrarily, e.g., zero in each case. If we knew and in (1.3d), then we could take and in (1.4d) and the solution would be identically , by the uniqueness of solutions of system (1.4), which will be shown below. The point, again, is that in many applications, we do not know and . We emphasize that in this algorithm, we construct our approximate solutions using only the observations of the velocity field solution, , in the -equation; no observations are needed for the temperature.
We will consider two types of interpolant observables. One is to be given by a linear interpolant operator satisfying the approximation property for every , where is a dimensionless constant. The other type is given by , together with for every , where are dimensionless constants. One example of an interpolant observable that satisfies (1.5) is the orthogonal projection onto the low Fourier modes with wave numbers such that . A more physical example is the volume elements that were studied in [9]. An example of an interpolant observable that satisfies (1.6) is given by the measurements at a discrete set of nodal points in (see Appendix A in [20]).
In the next section we lay out the functional setting commonly used in the mathematical study of the Navier–Stokes equations. We also recall the previous work on the Bénard problem establishing well-posedness and existence of a global attractor. In Section 3 we prove that solutions on the global attractor of (1.3) are determined by the velocity alone, a fact which motivates our data assimilation algorithm (1.4). The main results are in Section 3. Assuming adequate resolution in the observational data, and separately conditions (1.5), (1.6), we prove the well-posedness of system (1.4) as well as convergence (at an exponential rate) of the approximate solution of (1.4) to the reference solution of the Bénard convection problem (1.3).
Section snippets
Preliminaries
For the sake of completeness, this section presents some preliminary material and notation commonly used in the mathematical study of fluids, in particular in the study of the Navier–Stokes equations (NSE) and the Euler equations. For more detailed discussion on these topics, we refer the reader to [31], [32], [33], [34].
We begin by defining function spaces corresponding to the relevant physical boundary conditions. We define to be the set of functions defined in , which are
Convergence results
In this section, we derive conditions under which the approximate solution of the data assimilation system (3.4) converges to the solution of the Bénard convection problem (2.3) as . We will prove the result for observables operators that satisfy (2.6), (2.7), in functional settings, respectively.
The idea to apply data assimilation using observations of velocity only is inspired by the fact that solutions in the global attractor of (2.3) are completely determined by their
Discussion and final remarks
It is typical when implementing data assimilation to choose a relaxation parameter such as which is effective for the spatial resolution of the available data. The goal of our analysis, however, is to estimate in terms of physical parameters through rigorous bounds on the solutions in the global attractor. A sufficiently small value of is then determined in terms of . Thus, indirectly, the necessary spatial resolution depends on the physical parameters, which is natural.
We mention that
Acknowledgments
This work was completed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation (NSF). The work of A.F. is supported in part by NSF grant DMS-1418911. The work of M.S.J. is supported in part by NSF grants DMS-1008661, DMS-1109022 and DMS-1418911. The work of E.S.T. is supported in part by the NSF grants DMS-1009950, DMS-1109640 and DMS-1109645.
References (35)
- et al.
Determining finite volume elements for the 2D Navier–Stokes equations
Physica D
(1992) - et al.
Discrete data assimilation in the Lorenz and Navier–Stokes equations
Physica D
(2011) Data assimilation for the Navier–Stokes- equations
Physica D
(2009)- et al.
Attractors for the Bénard problem: existence and physical bounds on their fractal dimension
Nonlinear Anal. TMA
(1987) Global regularity for the 2D Boussinesq equations with partial viscosity terms
Adv. Math.
(2006)- et al.
Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid Voigt-regularization
J. Differential Equations
(2013) - et al.
Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems
Math. Comp.
(1997) - et al.
- et al.
Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension 2
Rend. Semin. Mat. Univ. Padova
(1967)
Asymptotic numerical analysis for the Navier–Stokes equations
Determination of the solutions of the Navier–Stokes equations by a set of nodal values
Math. Comp.
Determining nodes, finite difference schemes and inertial manifolds
Nonlinearity
Determining projections and functionals for weak solutions of the Navier–Stokes equations
Contemp. Math.
Upper bounds on the number of determining modes, nodes and volume elements for the Navier–Stokes equations
Indiana Univ. Math. J.
A numerical investigation of the interaction between the large scales and small scales of the two-dimensional incompressible Navier–Stokes equations, Research Report LA-UR-98–1712
Accuracy and stability of the continuous-times 3DVAR filter for the Navier–Stokes equations
Nonlinearity
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