Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations

Biswas, Animikh; Martinez, Vincent R.

doi:10.1016/j.nonrwa.2016.10.005

Cited by 44 publications

(33 citation statements)

References 28 publications

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“…The authors in [46] studied the convergence of the algorithm to the reference solution in the case of the two-dimensional subcritical surface quasi-geostrophic (SQG) equation. The convergence of this synchronization algorithm for the 2D NSE, in higher order (Gevery class) norm and in L ∞ norm, was later studied in [8]. An extension of the approach in [4] to the case when the observational data contains stochastic noise was analyzed in [7].…”

Section: B)mentioning

confidence: 99%

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

Farhat¹,

Lunasin²,

Titi³

2019

Partial Differential Equations Arising From Physics and Geometry

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In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary that one has to collect coarse mesh measurements of all the state variables, that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information of the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; viz. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model. This paper is dedicated to the memory of Professor Abbas Bahri MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.

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Section: B)mentioning

confidence: 99%

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

Farhat¹,

Lunasin²,

Titi³

2019

Partial Differential Equations Arising From Physics and Geometry

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“…For the incompressible 2D Navier-Stokes equations (NSE), it is shown in [4] that if µ is sufficiently large and h sufficiently small, specified explicitly in terms of the physical parameters, then v(t) − u(t) L 2 → 0 (or v(t) − u(t) H 1 → 0 under further assumptions on the size of µ), at an exponential rate (see also the computational work in [2,26]). The convergence of this synchronization algorithm for the 2D NSE, in higher order (Gevery class) norm and in L ∞ norm, was later studied in [7] for smoother forcing. An extension of the approach in [4] to the case when the observational data contains stochastic noise was analyzed in [6].…”

Section: Introductionmentioning

confidence: 99%

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

et al. 2018

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We introduce a continuous (downscaling) data assimilation algorithm for the 2D Bénard convection problem using vorticity or local circulation measurements only. In this algorithm, a nudging term is added to the vorticity equation to constrain the model. Our numerical results indicate that the approximate solution of the algorithm is converging to the unknown reference solution (vorticity and temperature) corresponding to the measurements of the 2D Bénard convection problem when only spatial coarse-grain measurements of vorticity are assimilated. Moreover, this convergence is realized using data which is much more coarse than the resolution needed to satisfy rigorous analytical estimates.

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“…Suppose that u is a solution of (2.11), corresponding to the initial value u 0 ∈ V . Then there exists a time t 0 which depends on u 0 such that for all t ≥ t 0 , it 8 holds that…”

Section: )mentioning

confidence: 99%

Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

Carlson¹,

Hudson²,

Larios³

2020

SIAM J. Sci. Comput.

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We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

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Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations

Cited by 44 publications

References 28 publications

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

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