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Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data

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Pith's one-line read In two dimensions, a nudged Smagorinsky-Ladyzhenskaya LES model converges exponentially to the Navier-Stokes solution, up to an error that vanishes like the square root of the turbulence viscosity.

desk verdict The mismatch setup is genuinely new and the error bound is the right kind of result, but Theorem 4.1 as stated needs stronger regularity on u than it assumes. read the letter →

arxiv 2508.07492 v1 pith:BC23UXU7 submitted 2025-08-10 math.AP physics.flu-dyn

Adam Larios , Ali Pakzad , Nicholas White This is my paper
classification math.APphysics.flu-dyn MSC 35Q3076F6593E1135K5576D0535B40
keywords Navier-StokesequationsLargeEddySimulationcontinuousdataassimilationnudgingSmagorinskymodelLadyzhenskayamodel-observationmismatchsynchronizationerror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Large eddy simulation is cheap enough for real forecasting, but the data fed into it come from the real fluid, not from the LES model itself. The paper targets that mismatch: it proves that a nudging term injecting coarse observations from the Navier-Stokes equations into a Smagorinsky-Ladyzhenskaya LES model drives the LES state exponentially close to the true flow in two dimensions. Exact synchronization is impossible because the LES model carries an extra turbulent-viscosity term the true equations lack, so the theorem leaves an error floor of order $\bar{\nu}^{1/2}$. That floor shrinks to zero as $\bar{\nu}\to 0$, recovering the classical no-mismatch result. The practical consequence is that LES-based data assimilation can track the true flow up to a controlled, quantifiable modeling error; the paper's 2D and 3D simulations show the predicted exponential decay to a plateau.

What carries the argument

The argument runs on the error equation for $w=u-v$. Because the LES model has the closure term $\nabla\cdot(\bar{\nu}|\nabla v|^{p-2}\nabla v)$ while the reference Navier-Stokes system does not, the usual monotonicity of the $p$-Laplacian cannot be applied directly; the paper rewrites the term in terms of $\nabla u$ and $\nabla w$, splitting it into four mismatch terms. These are controlled with Hölder, Young, Ladyzhenskaya, and Agmon inequalities using $H^1$ and $H^3$ bounds on $u$, while the nudging term contributes damping $-\mu\|w\|^2$ and a small term absorbed into $\nu\|\nabla w\|^2$ when $2\mu c_0 h^2\le\nu$. The Grashof-number condition $\mu\ge 8\nu\lambda_1 G^2$ ensures that the ti

What would settle it

Take a 2D periodic reference flow that is a strong Navier-Stokes solution but lacks $H^3$ regularity (so $\nabla u\notin L^\infty$), run the nudged Smagorinsky LES with parameters satisfying the theorem, and measure the long-time $L^2$ error; if it does not settle at a level of order $\bar{\nu}^{1/2}$, or if the prefactor grows as the reference's higher regularity degrades, the stated hypothesis is insufficient. A simpler test: vary $\bar{\nu}$ over several decades at fixed $\mu,h$ and check whether the plateau scales as $\bar{\nu}^{1/2}$.

Watch

Extended reading notes

Core claim

The central result is Theorem 4.1. For $p\ge 5/2$ in a 2D periodic domain, if the nudging parameter satisfies $\mu \ge 8\nu\lambda_1 G^2$ and the observation resolution satisfies $2\mu c_0 h^2 \le \nu$, then the difference $w=u-v$ obeys $\|w\|^2 \le C_{u,\Omega}\bar{\nu}(1-e^{-t}) + \|w(0)\| e^{-t}$. Hence the nudged LES solution $v$ converges exponentially fast to the Navier-Stokes solution $u$, and its asymptotic $L^2$ error is at most order $\bar{\nu}^{1/2}$. The constant $C_{u,\Omega}$ depends on the $H^1$ and $H^3$ norms of the reference solution and on the domain. The paper also proves global well-posedness of the assimilated system for $p\ge 5/2$, and numerical experiments in a 2D ann

Load-bearing premise

The load-bearing premise is that the true Navier-Stokes flow is smoother than just a strong solution: its gradient must be bounded (in 2D, $H^3$ regularity), because the proof needs that to control the extra turbulent-viscosity mismatch; the theorem as stated does not list this assumption.

Editorial extensions

If this is right

  • In two dimensions, an LES assimilation run from arbitrary initial data synchronizes to the true Navier-Stokes solution exponentially once $\mu$ and $h$ meet the theorem's conditions; only the turbulence viscosity $\bar{\nu}$ sets the asymptotic accuracy.
  • The asymptotic $L^2$ error floor scales like $\bar{\nu}^{1/2}$, so reducing the closure viscosity improves tracking, and the classical exact-synchronization result is recovered in the limit $\bar{\nu}\to 0$.
  • The assimilated system is globally well-posed for $p\ge 5/2$, so the feedback term does not destroy the LES model's longtime stability even with arbitrary initial data.
  • The numerical experiments with the Smagorinsky parameter $p=3$ in physical-boundary and 3D periodic settings reproduce exponential decay to a plateau, supporting the 2D theorem's error-floor mechanism as the operative behavior in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the paper leaves implicit is that the explicit size of $C_{u,\Omega}$ in terms of $\|u\|_{H^3}$ would let practitioners predict how rough a reference flow can be before the $\bar{\nu}^{1/2}$ floor is dominated by regularity effects.
  • The same four-term mismatch decomposition should carry over to time-dependent or dynamically determined $\bar{\nu}$ (as in the Germano dynamic model), so the error-floor mechanism is testable beyond the constant-coefficient case.
  • The 3D numerical plateau suggests the 2D proof's mechanism may extend to any sufficiently regular 3D Navier-Stokes solution, although a theorem cannot be expected without extra regularity assumptions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies continuous data assimilation (nudging) for a Smagorinsky/Ladyzhenskaya-type LES model when the observational data is generated by the full Navier–Stokes equations (NSE), rather than by the LES model itself. The main theoretical result (Theorem 4.1) claims that in a 2D periodic setting, for p≥5/2 and under conditions μ≥8νλ1G^2 and 2μc0h^2≤ν, the L2 error ∥u−v∥ between the NSE solution u and the nudged LES solution v satisfies ∥u−v∥^2 ≤ C_{u,Ω}ν̄(1−e^{-t})+∥u(0)−v(0)∥e^{-t}, so that ∥u−v∥ ≤ C ν̄^{1/2} as t→∞. Global well-posedness of the assimilated system is stated as Theorem 3.1 but only proved formally. Numerical experiments in a 2D annulus with no-slip boundaries and in a 3D periodic domain are presented as qualitative corroboration of the analytical prediction.

Significance. If established under the stated hypotheses, the result would be a meaningful contribution to continuous data assimilation in the realistic setting where model and observations come from different PDE systems. The paper correctly identifies the key analytical difficulty—the non-monotone p-Laplacian mismatch term—and proposes a plausible decomposition into four controlled terms. Explicit conditions on μ and h are given, and the inclusion of 2D simulations with physical boundary conditions and 3D periodic simulations is a useful complement to the theory. However, the central theorem currently has a load-bearing regularity gap and an incomplete Gronwall step; the numerical evidence is only qualitative. With proper repairs, the paper would be a solid contribution; in its present form the main claim is not fully justified.

major comments (3)
  1. [Section 4, Theorem 4.1 and Eq. (4.1)] The proof of Terms III, IV, and V uses Agmon's inequality in the form ∥∇u∥_{L∞} ≤ C∥u∥_{H1}^{1/2}∥u∥_{H3}^{1/2} (see the estimates below (4.1)). The theorem only assumes u is a strong solution of the 2D NSE. Standard strong solutions satisfy u∈L∞(0,T;H1) ∩ L2(0,T;H2), not H3, and H3 regularity is not implied by f∈L2. Consequently the constant C_{u,Ω}, which depends on ∥u∥_{H3}, may not be finite under the stated hypotheses. The theorem must either add a hypothesis such as u∈L∞(0,T;H3) or ∇u∈L∞, or the estimates must be reworked using only H2 regularity. This gap is load-bearing for the central bound.
  2. [Section 4, Eq. (4.2) and final Gronwall step] The differential inequality d/dt∥w∥^2 + (μ − (2/ν)∥∇u∥^2)∥w∥^2 ≤ C_{u,Ω}ν̄ is not closed by the stated conditions. The condition μ≥8νλ1G^2 with T=1/(νλ1) only ensures that the time-average of the coefficient μ−(2/ν)∥∇u∥^2 is nonnegative; pointwise the coefficient can be negative. A uniform Gronwall argument requires a strictly positive average to obtain exponential decay, and the decay rate and constants must be tracked. Moreover, the displayed conclusion contains a dimensional inconsistency: the term ∥u(0)−v(0)∥e^{-t} should be ∥u(0)−v(0)∥^2 e^{-ct} for some rate c. As written, the conclusion does not follow from (4.2).
  3. [Section 3, Theorem 3.1] Global well-posedness is not actually proved. The text states that only a 'formal proof' is provided and that the Galerkin argument is omitted, with Remark 3.2 referring to [14] for an alternative. Additionally, the proof assumes u∈C([0,T];H1) to define f_μ, but this regularity is not listed among the hypotheses of Theorem 3.1 (which only assumes f∈L∞((0,∞);L2)). Since Theorem 4.1 relies on the global existence of v, Theorem 3.1 must either be proved with appropriate hypotheses or restated as a conditional result with a precise reference. This is a load-bearing gap in the manuscript's logical structure.
minor comments (4)
  1. [Theorem 4.1 statement] The statement should specify that the estimate holds for t≥t0 (as used in the proof) and should clarify the meaning of the exponential e^{-t} (nondimensional time or with an explicit rate).
  2. [Section 5.1] The 2D numerical experiment is performed in an annulus with no-slip boundary conditions, whereas Theorem 4.1 is proved for periodic boundary conditions. The text says the numerics 'confirm' the theorem; it should explicitly note that the boundary conditions differ from the theoretical setting and that the agreement is qualitative.
  3. [Section 5.1, Figure 3 (right)] The claimed scaling ∥u−v∥∼ν̄^{1/2} is not quantified; a fitted slope with confidence interval would make the corroboration meaningful. Without this, the numerical evidence is only qualitative.
  4. [Throughout] Several typos and notational inconsistencies appear: 'expoenentially' (§1.1), 'Gr¨onwall' (missing umlaut), 'with with periodic' (Theorem 4.1), and the stress tensor notation |∇u|^{p−2}_F with the Frobenius subscript introduced only later.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity; central synchronization bound is derived from the PDEs. Only a minor, non-load-bearing self-citation to [14] (co-authored by Pakzad) occurs, plus an independent regularity gap in the proof.

full rationale

Theorem 4.1's error estimate is not circular. The proof starts from the actual difference equation obtained by subtracting (1.3) and (1.4), decomposes the p-Laplacian mismatch into Terms I-V, and bounds them with Hölder/Ladyzhenskaya/Agmon inequalities plus the NSE bounds in Theorem 2.1; the final exponential bound is a Grönwall estimate. The O(bar-nu^{1/2}) asymptotic error is a genuine perturbation estimate in the inserted LES viscosity coefficient, not a restatement or fit of an input. The only self-citation involving an author is [14] (Cao-Giorgini-Jolly-Pakzad), used in Remark 3.2 only as an alternative well-posedness route ('For further details, we direct interested readers to [14]') and in Section 5.2 as a numerical validation; the main well-posedness and synchronization arguments do not rely on [14]. This is therefore a minor non-load-bearing self-citation, not circularity. Independently, the proof of Theorem 4.1 invokes Agmon-type bounds on ||grad u||_{L^infty} that require more regularity than the stated 'strong solution' hypothesis (e.g., u in H3), so C_{u,Omega} may not be finite under the stated assumptions; this is a correctness gap, not a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central result rests on classical 2D NSE estimates, the interpolation-observable assumption, standard Sobolev inequalities, and an implicit H3 regularity requirement for the reference solution. The well-posedness of the LES model is sketched via standard Galerkin arguments and is not fully detailed. No ad hoc fitted parameters or new physical entities are introduced.

assumptions (4)
  • standard math 2D NSE solution u satisfies Theorem 2.1 bounds: existence of t0 with ||u(t)||^2 <= 2*nu^2*G^2 and integral_t^{t+T} ||nabla u||^2 <= 2*(1+T*nu*lambda1)*nu*G^2.
    Quoted from classical references [36,71] and used at the end of the synchronization proof.
  • domain assumption Interpolation operator Ih satisfies (1.1): boundedness on L2 and approximation error c0*h*||nabla phi||.
    Standard AOT assumption on observation operators, invoked in Sections 3 and 4.
  • domain assumption The reference NSE solution u has finite H1 and H3 norms for t >= t0, in particular nabla u in L-infinity.
    Needed in Terms III and V of the synchronization proof via Agmon's inequality; not stated as a hypothesis in Theorem 4.1.
  • standard math Standard Sobolev inequalities (Poincare, Ladyzhenskaya, Agmon, Lebesgue interpolation) hold on the domain.
    Used throughout the estimates; collected in Section 2.

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Pith. "Pith review of Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data." pith.science (2026). https://pith.science/paper/BC23UXU7

@misc{pith2026250807492,
  author       = {Pith},
  title        = {Pith review of: Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BC23UXU7}},
  note         = {Machine review of arXiv:2508.07492}
}
abstract

In atmospheric and turbulent flow modeling, Large Eddy Simulation (LES) is often used to reduce computational cost, while observational data typically originates from the underlying physical system. Motivated by this setting, we study a continuous data assimilation (CDA) algorithm applied to a Smagorinsky/Ladyzhenskaya-type LES model, in which the observational data is generated from the full Navier--Stokes equations (NSE). In the two-dimensional setting, we establish global well-posedness of the assimilated system and prove exponential convergence to the true solution, up to an error of order $\bar{\nu}^{1/2}$, where $\bar{\nu}$ is the turbulence viscosity parameter. In addition to rigorous analysis in 2D, we provide numerical simulations in both 2D domains with physical boundary conditions and 3D periodic domains, demonstrating effective synchronization in these cases, and corroborating our theoretical predictions.

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Cited by 2 Pith papers

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Reference graph

Works this paper leans on

72 extracted references · 69 canonical work pages · cited by 1 Pith paper

  1. [14]

    Y. Cao, A. Giorgini, M. Jolly, and A. Pakzad, Continuous data assimilation for the 3D Ladyzhenskaya model: analysis and computations , Nonlinear Anal. Real World Appl. 68 (2022), Paper No. 103659, 29

    work page 2022

  2. [1]

    Aitzhan, S

    A. Aitzhan, S. Sammak, P. Givi, and A. G. Nouri, PeleLM–FDF Large Eddy Simulator of Turbulent Combustion, Combustion Theory and Modelling 27 (2023), no. 1, 1–18

    work page 2023

  3. [2]

    D´ ebora A. F. Albanez, Helena J. Nussenzveig Lopes, and Edriss S. Titi, Continuous data assimilation for the three-dimensional Navier–Stokes- α model, Asymptotic Anal. 97 (2016), no. 1-2, 139–164

    work page 2016

  4. [3]

    M. U. Altaf, E. S. Titi, T. Gebrael, O. M. Knio, L. Zhao, M. F. McCabe, and I. Hoteit, Downscaling the 2D B´ enard convection equations using continuous data assimilation, Comput. Geosci. 21 (2017), no. 3, 393–410

    work page 2017

  5. [4]

    R. A. Anthes, Data assimilation and initialization of hurricane prediction models , J. Atmos. Sci. 31 (1974), no. 3, 702–719. DA IN LES: HANDLING MODEL-OBSER V ATION MISMATCH 21

    work page 1974

  6. [5]

    Auroux and J

    D. Auroux and J. Blum, A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm, Nonlin. Processes Geophys. 15 (2008), 305–319

    work page 2008

  7. [6]

    Azouani, E

    A. Azouani, E. Olson, and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci. 24 (2014), no. 2, 277–304

    work page 2014

  8. [7]

    Azouani and E

    A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm, Evol. Equ. Control Theory 3 (2014), no. 4, 579–594, DOI 10.3934/eect.2014.3.579

    doi:10.3934/eect.2014.3.579 2014

Show all 72 references
  1. [8]

    Bessaih, E

    H. Bessaih, E. Olson, and E. S. Titi, Continuous data assimilation with stochastically noisy data , Non- linearity 28 (2015), no. 3, 729–753

    2015

  2. [9]

    Biswas, Z

    A. Biswas, Z. Bradshaw, and M. Jolly, Data assimilation for the Navier-Stokes equations using local observables, SIAM J. Appl. Dyn. Sys. 20 (2021), no. 4, 2174–2203

    2021

  3. [10]

    Biswas, J Hudson, A

    A. Biswas, J Hudson, A. Larios, and Y. Pei, Continuous data assimilation for the 2D magnetohydrody- namic equations using one component of the velocity and magnetic fields , Asymptot. Anal. 108 (2018), no. 1-2, 1–43

    2018

  4. [11]

    Biswas and R

    A. Biswas and R. Price, Continuous data assimilation for the three-dimensional Navier–Stokes equations, SIAM J. Math. Anal. 53 (2021), no. 6, 6697–6723

    2021

  5. [12]

    Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011

    H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011

    2011

  6. [13]

    Br¨ ocker and I

    J. Br¨ ocker and I. G. Szendro,Sensitivity and out-of-sample error in continuous time data assimilation , Q. J. R. Meteorol. Soc. 138 (2012), no. 664, 785–801

    2012

  7. [15]

    Carlson, J

    E. Carlson, J. Hudson, A. Larios, V. R. Martinez, E. Ng, and J. P. Whitehead, Dynamically learning the parameters of a chaotic system using partial observations , Discrete Contin. Dyn. Syst. 42 (2022), no. 8, 3809–3839, DOI 10.3934/dcds.2022033

    2022 doi

  8. [16]

    Carlson, A

    E. Carlson, A. Larios, and E. S. Titi, Super-exponential convergence rate of a nonlinear continuous data assimilation algorithm: The 2D Navier–Stokes equations paradigm , J. Nonlin. Sci. 34 (2024), no. 2, 37, DOI 10.1007/s00332-024-10014-w

    2024 doi

  9. [17]

    Charney, M

    J. Charney, M. Halem, and R. Jastrow, Use of incomplete historical data to infer the present state of the atmosphere, J. Atmos. Sci. 26 (1969), 1160-1163

    1969

  10. [18]

    10, 103123, DOI 10.1063/5.0057421

    Nan Chen, Yuchen Li, and Evelyn Lunasin, An efficient continuous data assimilation algorithm for the Sabra shell model of turbulence , Chaos 31 (2021), no. 10, 103123, DOI 10.1063/5.0057421

    2021 doi

  11. [19]

    Y. T. Chow, W. T. Leung, and A. Pakzad, Continuous data assimilation for two-phase flow: analysis and simulations , J. Comput. Phys. 466 (2022), Paper No. 111395, 25

    2022

  12. [20]

    C ¸ ıbık, R

    A. C ¸ ıbık, R. Fang, W. Layton, and F. Siddiqua, Adaptive Parameter Selection in Nudging-Based Data Assimilation, Computer Methods in Applied Mechanics and Engineering 433 (2025), 117526

    2025

  13. [21]

    C ¸ ıbık, R

    A. C ¸ ıbık, R. Fang, W. Layton, and F. Siddiqua, Data Assimilation with Model Errors , arXiv preprint (2025), available at ����������

    2025

  14. [22]

    Clark Di Leoni, A

    P. Clark Di Leoni, A. Mazzino, and L. Biferale, Inferring flow parameters and turbulent configuration with physics-informed data assimilation and spectral nudging, Phys. Rev. Fluids 3 (2018), no. 10, 104604

    2018

  15. [23]

    Clark Di Leoni, A

    P. Clark Di Leoni, A. Mazzino, and L. Biferale, Synchronization to big data: Nudging the Navier–Stokes equations for data assimilation of turbulent flows , Phys. Rev. X 10 (2020), no. 1, 011023. 22 ADAM LARIOS, ALI PAKZAD, AND NICHOLAS WHITE

    2020

  16. [24]

    Y. J. Du and M. Shiue, Analysis and computation of continuous data assimilation algorithms for Lorenz 63 system based on nonlinear nudging techniques , J. Computat. Appl. Math. 386 (2021), 113246, DOI 10.1016/j.cam.2020.113246

    2021

  17. [25]

    Farhat, N

    A. Farhat, N. E. Glatt-Holtz, V. R. Martinez, S. A. McQuarrie, and J. P. Whitehead, Data assimilation in large Prandtl Rayleigh-B´ enard convection from thermal measurements, SIAM J. Appl. Dyn. Syst. 19 (2020), no. 1, 510–540

    2020

  18. [26]

    Farhat, H

    A. Farhat, H. Johnston, M. Jolly, and E. S. Titi, Assimilation of nearly turbulent Rayleigh-B´ enard flow through vorticity or local circulation measurements: a computational study , J. Sci. Comput. 77 (2018), no. 3, 1519–1533

    2018

  19. [28]

    Farhat, A

    A. Farhat, A. Larios, V. R. Martinez, and J. P. Whitehead, Identifying the body force from partial observations of a two-dimensional incompressible velocity field, Phys. Rev. Fluids 9 (2024), no. 5, 054602

    2024

  20. [29]

    Farhat, E

    A. Farhat, E. Lunasin, and E. S. Titi, Continuous data assimilation for a 2D B´ enard convection system through horizontal velocity measurements alone , J. Nonlinear Sci. 27 (2017), no. 3, 1065–1087

    2017

  21. [30]

    Farhat, E

    A. Farhat, E. Lunasin, and E. S. Titi, Data assimilation algorithm for 3D B´ enard convection in porous media employing only temperature measurements , J. Math. Anal. Appl. 438 (2016), no. 1, 492–506

    2016

  22. [31]

    Farhat, M

    A. Farhat, M. Jolly, and E. S. Titi, Continuous data assimilation for the 2D B´ enard convection through velocity measurements alone, Phys. D 303 (2015), 59–66

    2015

  23. [32]

    Farhat, E

    A. Farhat, E. Lunasin, and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field , J. Math. Fluid Mech. 18 (2016), no. 1, 1–23

    2016

  24. [33]

    Foia¸ s and G

    C. Foia¸ s and G. Prodi, Sur le comportement global des solutions non-stationnaires des ´ equations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1–34

    1967

  25. [34]

    Foias, C

    C. Foias, C. F. Mondaini, and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics , SIAM J. Appl. Dyn. Syst. 15 (2016), no. 4, 2109–2142

    2016

  26. [35]

    Frisch, Turbulence, Cambridge University Press, Cambridge, 1995

    U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995. The Legacy of A. N. Kolmogorov

    1995

  27. [36]

    Foias, O

    C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence , Encyclopedia of Mathematics and its Applications, vol. 83, Cambridge University Press, Cambridge, 2001

    2001

  28. [37]

    Garc ´ ıa-Archilla, J

    B. Garc ´ ıa-Archilla, J. Novo, and E. S. Titi,Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations , SIAM J. Numer. Anal. 58 (2020), no. 1, 410–429

    2020

  29. [38]

    Germano, U

    M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model , Physics of Fluids A 3 (1991), 1760–1765

    1991

  30. [39]

    Gesho, E

    M. Gesho, E. Olson, and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations , Commun. Comput. Phys. 19 (2016), no. 4, 1094–1110

    2016

  31. [40]

    Hecht, New development in FreeFem++, J

    F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265

    2012

  32. [41]

    J. E. Hoke and R. A. Anthes, The initialization of numerical models by a dynamic-initialization technique, Mon. Weather Rev. 104 (1976), no. 12, 1551–1556

    1976

  33. [42]

    Hudson and M

    J. Hudson and M. Jolly, Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations, J. Comput. Dyn. 6 (2019), no. 1, 131–145. DA IN LES: HANDLING MODEL-OBSER V ATION MISMATCH 23

    2019

  34. [43]

    H. A. Ibdah, C. F. Mondaini, and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates , IMA J. Numer. Anal. 40 (2020), no. 4, 2584–2625

    2020

  35. [44]

    John, Large Eddy Simulation of Turbulent Incompressible Flows , Lecture Notes in Computational Science and Engineering, vol

    V. John, Large Eddy Simulation of Turbulent Incompressible Flows , Lecture Notes in Computational Science and Engineering, vol. 34, Springer-Verlag, Berlin, 2004

    2004

  36. [45]

    M. S. Jolly, V. R. Martinez, E. J. Olson, and E. S. Titi, Continuous data assimilation with blurred-in- time measurements of the surface quasi-geostrophic equation , Chin. Ann. Math. Ser. B 40 (2019), no. 5, 721–764

    2019

  37. [46]

    M. S. Jolly and A. Pakzad, Data assimilation with higher order finite element interpolants , Internat. J. Numer. Methods Fluids 95 (2023), no. 3, 472–490

    2023

  38. [47]

    Jolly, T

    M. Jolly, T. Sadigov, and E. S. Titi, Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case, Nonlinear Anal. Real World Appl. 36 (2017), 287–317

    2017

  39. [48]

    M. S. Jolly, V. R. Martinez, and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud. 17 (2017), no. 1, 167–192

    2017

  40. [49]

    R. E. Kalman, A new approach to linear filtering and prediction problems , J. Basic Eng. 82 (1960), no. 1, 35–45

    1960

  41. [50]

    Kalnay, Atmospheric modeling, data assimilation, and predictability , Cambridge Univ

    E. Kalnay, Atmospheric modeling, data assimilation, and predictability , Cambridge Univ. Pr., 2003

    2003

  42. [51]

    O. A. Ladyzhenskaja, Modifications of the Navier–Stokes equations for large gradients of the velocities , Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)7 (1968), 126–154

    1968

  43. [52]

    O. A. Ladyzhenskaya, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems , Trudy Mat. Inst. Steklov. 102 (1967), 85–104

    1967

  44. [53]

    Lakshmivarahan and John M

    S. Lakshmivarahan and John M. Lewis, Nudging methods: A critical overview , Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, Vol. II, Springer, 2013, pp. 27–57

    2013

  45. [54]

    Larios, L

    A. Larios, L. Rebholz, and C. Zerfas, Global in time stability and accuracy of IMEX-FEM data assimila- tion schemes for Navier-Stokes equations , Comput. Methods Appl. Mech. Engrg. 345 (2019), 1077–1093

    2019

  46. [55]

    Larios and Y

    A. Larios and Y. Pei, Approximate continuous data assimilation of the 2D Navier–Stokes equations via the Voigt-regularization with observable data , Evol. Equ. Control Theory 9 (2020), no. 3, 733–751, DOI 10.3934/eect.2020031

    2020 doi

  47. [56]

    Larios and Y

    A. Larios and Y. Pei, Nonlinear continuous data assimilation , Evol. Equ. Control Theory 13 (2024), no. 2, 329–348, DOI 10.3934/eect.2023048

    2024 doi

  48. [57]

    Larios and C

    A. Larios and C. Victor, The second-best way to do sparse-in-time continuous data assimilation: Im- proving convergence rates for the 2D and 3D Navier–Stokes equations , available at ��������������� . 25 pp. (submitted)

  49. [58]

    K. Law, A. Stuart, and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, vol. 62, Springer, Cham, 2015

    2015

  50. [59]

    Layton, Introduction to the numerical analysis of incompressible viscous flows, Computational Science & Engineering, vol

    W. Layton, Introduction to the numerical analysis of incompressible viscous flows, Computational Science & Engineering, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008

    2008

  51. [60]

    D. K. Lilly, The representation of small-scale turbulence in numerical simulation experiments , IBM Scientific Computing Symposium on Environmental Sciences, New York, USA (1967), 195–210

    1967

  52. [61]

    Lunasin and E

    E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems—a computational study , Evol. Equ. Control Theory 6 (2017), no. 4, 535–557

    2017

  53. [62]

    P. A. Markowich, E. S. Titi, and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity 29 (2016), no. 4, 1292–1328. 24 ADAM LARIOS, ALI PAKZAD, AND NICHOLAS WHITE

    2016

  54. [63]

    C. F. Mondaini and E. S. Titi, Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm , SIAM J. Numer. Anal. 56 (2018), no. 1, 78–110

    2018

  55. [64]

    Newey, J

    J. Newey, J. P. Whitehead, and E. Carlson,Model discovery on the fly using continuous data assimilation, arXiv e-prints (2024). arxiv2411.13561

    2024 arXiv

  56. [65]

    Pawar, S

    S. Pawar, S. E. Ahmed, O. San, A. Rasheed, and I. M. Navon, Long short-term memory embedded nudging schemes for nonlinear data assimilation of geophysical flows , Physics of Fluids 32 (2020)

    2020

  57. [66]

    Pei, Continuous data assimilation for the 3D primitive equations of the ocean , Commun

    Y. Pei, Continuous data assimilation for the 3D primitive equations of the ocean , Commun. Pure Appl. Anal. 18 (2019), no. 2, 643–661

    2019

  58. [67]

    S. B. Pope, Ten questions concerning the large-eddy simulation of turbulent flows , New J. Phys. 18 (2004), no. 6

    2004

  59. [68]

    S. B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000

    2000

  60. [69]

    Sharan, G

    N. Sharan, G. Matheou, and P. E. Dimotakis, Turbulent shear-layer mixing: initial conditions, and direct-numerical and large-eddy simulations, Journal of Fluid Mechanics 880 (2019), 1005–1042

    2019

  61. [70]

    Smagorinsky, General circulation experiments with the primitive equations

    J. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment , Mon. Weather Rev. 91 (1963), 99–164

    1963

  62. [71]

    Temam, Navier-Stokes equations

    R. Temam, Navier-Stokes equations. Theory and numerical analysis , Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977

    1977

  63. [72]

    D. C. Wilcox, Turbulence Modeling for CFD , 3rd ed., DCW Industries, La Canada, CA, 2006

    2006

  64. [73]

    Zerfas, L

    C. Zerfas, L. G. Rebholz, M. Schneier, and T. Iliescu, Continuous data assimilation reduced order models of fluid flow , Comput. Methods Appl. Mech. Engrg. 357 (2019), 112596, 18. (Adam Larios) Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, ...

    2019

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