A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Gesho, Masakazu; Olson, Eric J.; Titi, Edriss S.

doi:10.4208/cicp.060515.161115a

Cited by 91 publications

(91 citation statements)

References 19 publications

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“…In the previous sections, we showed that the data assimilation algorithm (2.13) can still perform well even when there is error in the viscosity parameter, provided that µ is large and h is small. However, for large values of the Grashof number satisfying the requirements of the rigorous estimates would require prohibitively small values of h. Fortunately, in practice the requirements on h and µ need not be strict when Re 1 is known, and in fact we would expect the algorithm to perform well with very modest values for h and µ (see [31]).…”

Section: Numerical Resultsmentioning

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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Section: Numerical Resultsmentioning

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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“…Moreover, the algorithm may continue to work with values of h much larger and values of λ much smaller than required by our analysis. For example, computational experiments performed by [16] for a different spatially filtered continuous data assimilation method based on nudging show that the method performs far better than the analytical estimates suggest. We conjecture similar numerical effectiveness for the discrete data assimilation method described in the present paper.…”

Section: Discussionmentioning

confidence: 99%

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Çelik¹,

Olson²,

Titi³

2019

SIAM J. Appl. Dyn. Syst.

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We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.and let Q λ = I − P λ be the orthogonal complement of P λ . Now, given λ > 0 and I h define J = P λ P σ I h and E = I − J.(1.6) Note, although no additional orthogonality or regularity properties other than those appearing in Definition 1.1 have been assumed on I h , the above spectral filtering yields an operator J which is nearly orthogonal and has a range contained in D(A). The downscaling data assimilation algorithm studied in this paper may now be stated as Definition 1.2. Let U be an exact solution of (1.2) which evolves according to dynamics given by the semi-process S. Let t n = t 0 + nδ be a sequence of times for which partial

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“…Regarding related numerical studies, [GOT16] demonstrated in the case of the 2D NSE that the number of observables required for synchronization using (1.2) is much lower in practice than what has been deemed sufficient by the rigorous analysis. In the setting of the 2D RB system, numerical studies were carried out in [ATG + 17], and then in [FJJT18] for nearly turbulent flows using vorticity and local circulation measurements.…”

Section: Introductionmentioning

confidence: 99%

Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

Farhat¹,

Glatt-Holtz²,

Martinez³

et al. 2020

SIAM J. Appl. Dyn. Syst.

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This work applies a continuous data assimilation scheme-a particular framework for reconciling sparse and potentially noisy observations to a mathematical model-to Rayleigh-Bénard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to the earth's mantle and to gases under high pressure. We rigorously identify conditions that guarantee synchronization between the observed system and the model, then confirm the applicability of these results via numerical simulations. Our numerical experiments show that the analytically derived conditions for synchronization are far from sharp; that is, synchronization often occurs even when the conditions of our theorems are not met. We also develop estimates on the convergence of an infinite Prandtl model to a large (but finite) Prandtl number generated set of observations. Numerical simulations in this hybrid setting indicate that the mathematically rigorous results are accurate, but of practical interest only for extremely large Prandtl numbers.

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A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Cited by 91 publications

References 19 publications

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

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

Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

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