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Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that for the Smagorinsky and the new Ladyzhenskaya reduced order models, the ROM error is bounded by the closure error and that both models are limit consistent.

desk verdict New L-ROM and a first ROM-level limit-consistency framework with clean proofs, but the verifiability theorem carries an unquantified commutation-error gap. read the letter →

arxiv 2505.18310 v1 pith:RA7AXDMU submitted 2025-05-23 physics.flu-dyn cs.NAmath.NA

Jorge Reyes , Ping-Hsuan Tsai , Ian Moore , Honghu Liu , Traian Iliescu This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA MSC 65M1565M6076D0576F65
keywords reducedordermodellargeeddysimulationviscosityclosureSmagorinskyROMLadyzhenskayaverifiabilitylimitconsistencyproperorthogonaldecomposition
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

The paper asks when an eddy-viscosity closure for a reduced order model (ROM) can be trusted in under-resolved, convection-dominated flows. It introduces the Ladyzhenskaya ROM (L-ROM), a generalization of the Smagorinsky ROM, and proves two properties for both models. Verifiability: the ROM error is bounded, up to a constant, by the closure error, which is the difference between the exact subfilter-scale stress and the modeled one. Limit consistency: as the ROM dimension approaches the rank of the snapshot data and the ROM lengthscale goes to zero, the ROM solution converges to the solution of the d-dimensional Galerkin ROM. These are, to the authors' knowledge, the first verifiability proof for eddy-viscosity ROMs and the first limit-consistency result at the ROM level, and they are confirmed numerically on the 1D Burgers equation and the 2D lid-driven cavity at Re = 15,000.

What carries the argument

The machinery is a discrete LES-ROM construction built on the ROM L2 projection $P_r$ as the spatial filter. Filtering the d-dimensional Galerkin ROM produces an exact subfilter-scale stress tensor $\tau_{\mathrm{FOM}}(u_d) = P_r(u_d \cdot \nabla u_d) - P_r(u_d) \cdot \nabla P_r(u_d)$ and a commutation error $E^n$ that the paper sets to zero, following earlier work. The closure $\tau_{\mathrm{ROM}}$ for the L-ROM is the eddy-viscosity operator $-(C_S\delta)^\mu \nabla\cdot(\|\nabla w_r\|_F^s \nabla w_r)$, with the S-ROM the special case $\mu = 2$, $s = 1$. The proof engine is a stability estimate: subtract the LES-ROM from the filtered d-dimensional ROM, test with the error, use mean dissipativity (from strong monotonicity, Lemma 2) to drop the closure difference term, bound the trilinear term via Lemma 1, and close with the discrete Gronwall inequality (Lemma 3). For limit consistency, Lemma 5 sends $\delta \to 0$ at fixed $r$, Lemma 6 bounds the G-ROM gap as $r \to d$, and Theorem 2 combines them by the triangle inequality.

What would settle it

Compute the discrete commutation-error norm $|||P_r(\nabla u_d) - \nabla P_r(u_d)|||$ on the same snapshots used in the Burgers and cavity tests and compare it with the closure residual $|||P_r(\tau_{\mathrm{FOM}}(u_d) - \tau_{\mathrm{ROM}}(P_r(u_d)))|||$. If the commutation term is not much smaller, the theorem's right-hand side omits a term of comparable size and the analysis does not apply to the solved equations; a second check is to run the cavity tests with backward Euler instead of BDF3/EXT3 and see whether the claimed rate survives.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a pair of a priori error bounds. Theorem 1 shows that any LES-ROM whose closure satisfies a mean dissipativity condition obeys $|||P_r(u_d) - w_r|||^2_{\infty,0} + Re^{-1}|||\nabla(P_r(u_d) - w_r)|||^2_{2,0} \leq K\,|||P_r(\tau_{\mathrm{FOM}}(u_d) - \tau_{\mathrm{ROM}}(P_r(u_d)))|||^2_{2,0}$, so the ROM error is controlled by the computable closure residual. Corollaries 1.1 and 1.2 verify the condition for the L-ROM and S-ROM via the strong monotonicity of the Ladyzhenskaya operator. Theorem 2 then bounds $|||u_d - w_r|||$ by the projection error plus the closure term, and Theorem 3 shows this bound goes to zero as $r \to d$ and $\delta \to 0$, making both models limit consistent in the discrete sense the paper defines. Numerical experiments on the Burgers equation and the Re = 15,000 lid-driven cavity reproduce the predicted linear scaling between ROM error and closure error.

Load-bearing premise

The load-bearing premise is that the commutation error $E^n = P_r(\nabla u_d) - \nabla P_r(u_d)$ is negligible, so the filtered equation analyzed in the theorems is exactly the equation solved numerically; the paper also assumes backward-Euler stability arguments carry over to the BDF3/EXT3 time stepping used in the cavity test.

Editorial extensions

If this is right

  • Any eddy-viscosity closure that satisfies the mean dissipativity condition (25) inherits the verifiability bound of Theorem 1, so the result covers more than the two models tested.
  • Because the right-hand side of the bound is built from computable quantities (the projection of the exact and modeled stresses), verifiability supplies a practical, a posteriori error indicator for the ROM.
  • For the L-ROM and S-ROM, the limit-consistency bound of Theorem 2 implies convergence to the d-dimensional Galerkin solution as $r \to d$ and $\delta \to 0$, with rates $O(\delta^\mu)$ and $O(\delta^2)$ respectively.
  • The numerical experiments reproduce the predicted log-log slope near 1 between ROM error and closure error for both the Burgers and cavity problems, so the theory is consistent with computation.

Reading between the lines

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

  • Because the bound's constant grows exponentially in $Re^3$, the practical guarantee weakens sharply at very high Reynolds numbers; the theory is most informative in the moderate-Re under-resolved regime.
  • One could test the commutation-error premise directly: if a run with a differential filter (where $P_r$ is replaced by a smooth filter) changes the measured slope, the neglected term is not actually harmless for those parameters.
  • Coupling the lengthscale $\delta$ to the ROM dimension $r$, as the paper suggests via energy-based lengthscales, would reduce the two-parameter limit-consistency statement to a single-parameter one and may give sharper rates.
  • The verifiability framework likely transfers to other mean-dissipative closures, such as variational multiscale or data-driven eddy-viscosity models, since Theorem 1 does not use the specific form of $\tau_{\mathrm{ROM}}$ beyond monotonicity.
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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 / 6 minor

Summary. The manuscript proposes a Ladyzhenskaya ROM (L-ROM), an eddy-viscosity closure generalizing the Smagorinsky ROM (S-ROM), and analyzes both within an LES-ROM framework based on the ROM L2 projection. Section 3 defines verifiability and mean dissipativity, proves that any mean-dissipative LES-ROM is verifiable (Theorem 1), and specializes this to the L-ROM and S-ROM (Corollaries 1.1 and 1.2). Section 4 introduces a discrete notion of ROM limit consistency, proves a fixed-r bound between the LES-ROM and the r-dimensional G-ROM as the lengthscale δ goes to zero (Lemma 5), a bound between r- and d-dimensional G-ROMs (Lemma 6), and combines these to show that the L-ROM and S-ROM are limit consistent as r approaches d and δ goes to zero (Theorems 2 and 3). Numerical experiments in 1D Burgers (ν=2×10^-3) and 2D lid-driven cavity (Re=15,000) display the predicted error-versus-closure-error scaling and δ-convergence rates.

Significance. If made fully rigorous, the paper would provide the first verifiability and discrete limit-consistency results for eddy-viscosity LES-ROMs, and the proposed L-ROM is a useful tunable generalization of the S-ROM. The main proofs are self-contained and use standard tools (trilinear estimates, strong monotonicity, discrete Gronwall) without parameter fitting; the numerical experiments cover genuinely convection-dominated settings and report rates consistent with the theory, e.g., O(δ^{2μ}) for the closure term. However, the significance is currently qualified by two gaps: the unquantified commutation-error neglect in the verifiability theorem, and the r-dependence of the Gronwall constant in the limit-consistency theorem. Both are fixable within the manuscript's scope, but they need to be addressed before the advertised claims are fully supported.

major comments (3)
  1. [Section 2.3 and Theorem 1 (Eqs. (20)-(27))] The filtered d-dimensional G-ROM (20) contains the commutation-error term Re^{-1}(E^n(u_d), ∇v_r) with E^n defined in (22). When (23) is subtracted from (20) and the resulting equation is tested with e^n, this term should appear in the error balance; the displayed error equation (27) contains no such term, and no bound for it is supplied. A standard estimate would contribute C Re^{-1}|||E|||^2_{2,0} to the right-hand side of (26), so the theorem's conclusion that the ROM error is bounded by the closure error alone is not established for the model (20)-(23). This is not merely cosmetic: at r=d one has E^n = P_d(∇u_d) - ∇P_d(u_d) = P_d(∇u_d) - ∇u_d, which need not vanish because ∇u_d is generally not in X_d. The paper's appeal to [25] for smallness at large Re is qualitative and is not reflected in any assumption in Theorem 1; the Burgers test at ν=2×10^-3 (Re≈500) is not an asymptotic large-Re regime. I note that Theorem 2's proof goes through Lemma 5 and does not use (20), so this gap affects Theorem 1 and Corollaries 1.1-1.2 rather than the limit-consistency theorem as proved.
  2. [Lemma 5 and Theorem 2 (Eqs. (51)-(53), (67))] In the proof of Lemma 5, the Gronwall coefficient arises from the bound of |b*(e^n,u_r^n,e^n)| and therefore contains ||∇u_r^n||^4, but the statement of Lemma 5 and the final display set d_n = 1 + C Re^3 ||∇u_d^n||^4. This is a mismatch (likely a typo). If d_n depends on u_r, then the exponential factor K in (53) and in the Theorem 2 bound (67) depends on r through sup_n ||∇u_r^n||^4. No uniform-in-r bound for this quantity is proved, so the statement in Theorem 2 that K depends only exponentially on Re^3 is not justified, and condition (1) of Definition 4.1 (BLC decreases as r increases) does not follow from the displayed bound. The authors should correct the typo, add an explicit uniform-in-r regularity assumption, or weaken the claim accordingly.
  3. [Remark 3 and Section 5.4] The numerical cavity experiments use the BDF3/EXT3 time-stepping scheme, while Theorems 1-3 are proved for backward Euler. Remark 3 and Section 5.2 state the extension to higher-order schemes as a belief rather than a proof. Consequently, Figs. 8-11 demonstrate the predicted scalings for a discretization not covered by the theorems, so the numerical validation is weaker than the text suggests. The authors should either prove the extension, rerun the cavity tests with backward Euler, or clearly label the BDF3/EXT3 results as heuristic evidence.
minor comments (6)
  1. [Lemma 3 and Theorems 1-2] Lemma 3 requires Δt d_n < 1, but Theorem 1, Corollaries 1.1-1.2, Lemma 5, and Lemma 6 only assume Δt d_n ≤ 1. The strict inequality should be used, or a discrete Gronwall variant allowing equality should be cited.
  2. [Lemma 5 statement] In Lemma 5, the coefficient d_n is written as 1 + C Re^3 ||∇u_d^n||^4, while the proof just before (51) has ||∇u_r^n||^4; please correct this inconsistency (see also Major Comment 2).
  3. [Section 5.3.2] The text says 'we take a sequence {(r_i,δ_i)}^8_{i=1}' but lists only seven pairs; correct the count or add the missing pair.
  4. [Section 5.4.1 and Fig. 9] The L-ROM linear-regression slope in Fig. 9(b) is 1.53, which does not match the stated criterion α≈1 in (73); the text should either explain this discrepancy or adjust the criterion.
  5. [Remark 8] Remark 8 replaces the d-dimensional G-ROM solution u_d by P_d(u_FOM) in the numerical metrics; since these quantities are not identical, the numerical errors are not exactly the quantities in Theorems 1-2. This approximation should be stated explicitly in the interpretation of the numerical results.
  6. [Section 1] Spelling: 'Burger equation' should be 'Burgers equation'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; proofs are self-contained conditional on a self-cited commutation-error assumption.

full rationale

After walking the chain from the filtered d-dimensional G-ROM (20) through Theorem 1, Corollaries 1.1-1.2, Lemmas 5-6, and Theorems 2-3, I find no step in which a derived quantity is identical by construction to an input, nor any fitted parameter renamed as a prediction. Theorem 1 is a standard energy estimate: subtracting the LES-ROM (23) from (20), using skew-symmetry and mean dissipativity, and applying Gronwall yields exactly the closure-error bound (26). Corollaries 1.1 and 1.2 verify mean dissipativity through the strong monotonicity lemma. Lemma 5 shows w_r to u_r as delta goes to 0 because the closure residual enters the right-hand side and Assumption 1 makes it vanish; Theorem 3 verifies Assumption 1 from the explicit (C_S delta)^mu factor. Lemma 6 shows u_r to u_d as r approaches d through the POD projection error eta, which is the standard hierarchy convergence rather than an assumed conclusion. The limit-consistency result is 'by construction' only in the benign sense that the closure operator explicitly vanishes as delta goes to 0, which is exactly what consistency means. The one caveat is the neglect of the commutation error E^n following self-citations [25] and [26]: equation (27) drops a term present in (20), and no quantitative bound for E^n is supplied. This is an unverified modeling assumption and a genuine correctness risk, but it is not circular: the theorem is proved for the commutation-error-free equation, and the algebra does not assume the target bound. The numerical use of BDF3/EXT3 after a backward-Euler analysis is also a stated limitation (Remark 3), not a circular step. Overall, the central derivations are self-contained once the stated assumptions are granted, so the circularity score is low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central theorems do not rely on fitted constants; C_S, s, mu, and delta are tunable model and numerical parameters. The main modeling axioms are the weakly divergence-free ROM setting, neglect of the commutation error, and the backward Euler assumption that is then relaxed in the numerical tests.

free parameters (3)
  • Smagorinsky constant C_S = 1
    Set to 1 in both S-ROM and L-ROM implementations (Section 5.2). It is a tunable model constant, though the theorems hold for any positive value.
  • L-ROM exponents (s, mu) = s=2, mu=10/3 for L-ROM; s=1, mu=2 for S-ROM
    Chosen from existing Ladyzhenskaya FOM results [16] and from the Smagorinsky model definition (Section 5.2). These choices set the displayed convergence rates in Figures 6 and 11.
  • ROM lengthscale delta = Various: 0.001, 0.04, and geometric sequences down to 1e-5
    Selected per experiment: delta = h = 0.001 for Burgers verifiability, delta = 0.001 for cavity verifiability, and hand-chosen decreasing sequences for limit consistency (Sections 5.3 and 5.4). The proofs are valid for any delta.
assumptions (4)
  • standard math Standard functional analysis bounds: Lemma 1 (trilinear form estimates), Lemma 2 (strong monotonicity), Lemma 3 (discrete Gronwall).
    Used in the proofs of Theorems 1-3 and Lemmas 5-6; cited from [31,52,13,33,35,15].
  • domain assumption The ROM velocity is weakly divergence free and inherits homogeneous Dirichlet boundary conditions; the skew-symmetric trilinear form b* is used for stability.
    Remark 1 and Section 2.2; all ROM spaces X_r are subspaces of X = W^{1,2}_0(Omega).
  • domain assumption The commutation error E^n(u_d) = P_r(nabla u_d) - nabla P_r(u_d) is neglected in the filtered d-dimensional G-ROM.
    Assumed in Section 2.3 after equation (22), following [26]; the paper notes E^n is generally nonzero but negligible for large Re.
  • domain assumption Backward Euler time discretization is used in all proofs; higher-order time stepping is assumed extendable.
    Remark 3 states the belief that results extend to consistent higher-order time discretizations, and Section 5.2 uses BDF3/EXT3 for the cavity, so the numerical validation relies on this extension.

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Pith. "Pith review of Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models." pith.science (2026). https://pith.science/paper/RA7AXDMU

@misc{pith2026250518310,
  author       = {Pith},
  title        = {Pith review of: Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RA7AXDMU}},
  note         = {Machine review of arXiv:2505.18310}
}
abstract

Large eddy simulation reduced order models (LES-ROMs) are ROMs that leverage LES ideas (e.g., filtering and closure modeling) to construct accurate and efficient ROMs for convection-dominated (e.g., turbulent) flows. Eddy viscosity (EV) ROMs (e.g., Smagorinsky ROM (S-ROM)) are LES-ROMs whose closure model consists of a diffusion-like operator in which the viscosity depends on the ROM velocity. We propose the Ladyzhenskaya ROM (L-ROM), which is a generalization of the S-ROM. Furthermore, we prove two fundamental numerical analysis results for the new L-ROM and the classical S-ROM: (i) We prove the verifiability of the L-ROM and S-ROM, i.e, that the ROM error is bounded (up to a constant) by the ROM closure error. (ii) We introduce the concept of ROM limit consistency (in a discrete sense), and prove that the L-ROM and S-ROM are limit consistent, i.e., that as the ROM dimension approaches the rank of the snapshot matrix, $d$, and the ROM lengthscale goes to zero, the ROM solution converges to the \emph{``true solution"}, i.e., the solution of the $d$-dimensional ROM. Finally, we illustrate numerically the verifiability and limit consistency of the new L-ROM and S-ROM in two under-resolved convection-dominated problems that display sharp gradients: (i) the 1D Burgers equation with a small diffusion coefficient; and (ii) the 2D lid-driven cavity flow at Reynolds number $Re=15,000$.

Figures

Figures reproduced from arXiv: 2505.18310 by the authors.

Figure 1
Figure 1. Schematic of limit consistency in the LES–ROM context. [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The Burgers equation at ν = 2 × 10−3 . Predicted value of u comparison between the G-ROM, S-ROM, and L-ROM at r = 10 with δ = 0.04. Other choices can (and probably should) be used in the ROM setting, e.g., ROM lengthscales that adapt as we change the ROM dimension (see the energy-based ROM lengthscale [36]). The results generally show that, as εclosure decreases, so does εROM for both the S-ROM and L-ROM [PITH_FULL… view at source ↗
Figure 3
Figure 3. The Burgers equation at ν = 2 × 10−3 . The ROM error εROM and the closure error εclosure with respect to different r values for the S-ROM and L-ROM with CS = 1 and δ = 0.001. (a) S-ROM with δ = h = 0.001 (b) L-ROM with δ = h = 0.001 [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The Burgers equation at ν = 2 × 10−3 . The behavior of the ROM error εROM with respect to the closure error εclosure. Values shown are for r = 15, . . . , 35. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]
Figure 5
Figure 5. Figure 5: The Burgers equation at ν = 2 × 10−3 , demonstrating the limit consis￾tency of the S-ROM and L-ROM. Finally, following Remark 7, equation (68) in Theorem 3 suggests that ∆t PM n=1 ∥τ ROM(u n r )∥ should scale like O(δ µ ). Recall that we have chosen µ = 10/3 for our im…
Figure 6
Figure 6. Figure 6: The Burgers equation at ν = 2 × 10−3 . The regression result demon￾strates the rate suggested by Theorem 3. 5.4. 2D Lid-Driven Cavity Our next test case is the 2D lid-driven cavity problem at Re = 15, 000. We consider the direct numerical simulation to be the FOM. A de…
Figure 7
Figure 7. Figure 7: 2D lid-driven cavity at Re = 15, 000. Predicted velocity magnitude comparison between the G-ROM, S-ROM, and L-ROM with δ = 0.006, along with the FOM velocity magnitude. mode, φ0 , is set to be the FOM velocity field at the initial time instance, i.e., at t = 6000. We a…
Figure 8
Figure 8. Figure 8: 2D lid-driven cavity at Re = 15, 000. The ROM error εROM and the closure error εclosure with respect to different r values for the S-ROM and L-ROM with CS = 1 and δ = 0.001. The results generally show that, as εclosure decreases, so does εROM for both the S-ROM and L-R…
Figure 9
Figure 9. Figure 9: 2D lid-driven cavity at Re = 15, 000. The behavior of the ROM error εROM with respect to the closure error εclosure. verifiability of the S-ROM and L-ROM for different time interval lengths that are less than 20. For lengths that are greater than 20, we found that the …
Figure 10
Figure 10. Figure 10: 2D lid-driven cavity at Re = 15, 000. The limit consistency of the S-ROM and L-ROM. the sequence {(8, 2.5×10−2 ),(10, 10−2 ),(12, 5×10−3 ),(16, 2.5×10−3 ),(20, 10−3 ), (24, 5×10−4 )} for the L-ROM, and plot in [PITH_FULL_IMAGE:figures/full_fig_p036_10.png]
Figure 11
Figure 11. Figure 11: 2D lid-driven cavity at Re = 15, 000. The regression results demonstrate the rate suggested by Theorem 3 and Remark 7. closure, consists of a diffusion-like operator with a viscosity that depends on the Frobenius norm of the gradient of the ROM velocity. For both the …

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Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [25]

    B. Koc, M. Mohebujjaman, C. Mou, and T. Iliescu, Commutation error in reduced order modeling of fluid flows , Adv. Comput. Math. 45 (2019), no. 5-6, 2587–2621

    work page 2019

  2. [1]

    S. D. Agdestein and B. Sanderse, Discretize first, filter next: Learning divergence- consistent closure models for large-eddy simulation , J. Comput. Phys. 522 (2025), 113577

    work page 2025

  3. [2]

    Ahmed and O

    M. Ahmed and O. San, Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks , Comp. Appl. Math. 37 (2018), no. 5, 6870–6902

    work page 2018

  4. [3]

    E Ahmed, S

    S. E Ahmed, S. Pawar, O. San, A. Rasheed, T. Iliescu, and B. R Noack, On closures for reduced order models—a spectrum of first-principle to machine-learned avenues , Phys. Fluids 33 (2021), no. 9

    work page 2021

  5. [4]

    Ballarin, T

    F. Ballarin, T. C. Rebollo, E. D. Avila, M. G. M´ armol, and G. Rozza,Certified reduced basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height, Comput. Math. Appl. 80 (2020), no. 5, 973–989

    work page 2020

  6. [5]

    H. Beir˜ ao da Veiga,On the regularity of flows with Ladyzhenskaya shear-dependent vis- cosity and slip or non-slip boundary conditions, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 58 (2005), no. 4, 552–577

    work page 2005

  7. [6]

    Bergmann, C

    M. Bergmann, C. H. Bruneau, and A. Iollo, Enablers for robust POD models , J. Comput. Phys. 228 (2009), no. 2, 516–538

    work page 2009

  8. [7]

    Berkooz, P

    G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows , Ann. Rev. Fluid Mech. 25 (1993), no. 1, 539–575

    work page 1993

Show all 60 references
  1. [8]

    L. C. Berselli, T. Iliescu, and W. J. Layton, Mathematics of large eddy simulation of turbulent flows , Scientific Computation, Springer-Verlag, Berlin, 2006. MR2185509 (2006h:76071)

    2006

  2. [9]

    Y. Cao, A. Giorgini, M. Jolly, and A. Pakzad, Continuous data assimilation for the 3D Ladyzhenskaya model: analysis and computations , Nonlinear Analysis: Real World Applications 68 (2022), 103659. 38

    2022

  3. [10]

    Chaturantabut and D

    S. Chaturantabut and D. C Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal. 50 (2012), no. 1, 46–63

    2012

  4. [11]

    Couplet, P

    M. Couplet, P. Sagaut, and C. Basdevant, Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow , J. Fluid Mech. 491 (2003), 275–284. MR2014314

    2003

  5. [12]

    Du and M

    Q. Du and M. D Gunzburger, Finite-element approximations of a Ladyzhenskaya model for stationary incompressible viscous flow , SIAM J. Numer. Anal. 27 (1990), no. 1, 1–19

    1990

  6. [13]

    , Analysis of a Ladyzhenskaya model for incompressible viscous flow , J. Math. Anal. Appl. 155 (1991), no. 1, 21–45

    1991

  7. [14]

    Germano, U

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

    1991

  8. [15]

    G Heywood and R

    J. G Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. Part IV: Error analysis for second-order time discretization , SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384

    1990

  9. [16]

    S. C. Huang, A. Johnson, M. Neda, J. Reyes, and H. Tehrani, A generalization of the Smagorinsky model, Appl. Math. Comput. 469 (2024), 128545

    2024

  10. [17]

    T. J. R. Hughes, G. R. Feij´ oo, L. Mazzei, and J.-B. Quincy,The variational multiscale method—a paradigm for computational mechanics , Comput. Methods Appl. Mech. Engrg. 166 (1998), no. 1, 3–24. Advances in Stabilized Methods in Computational Mechanics

    1998

  11. [18]

    Iliescu, H

    T. Iliescu, H. Liu, and X. Xie, Regularized reduced order models for a stochastic Burgers equation, International Journal of Numerical Analysis & Modeling 15 (2018), 594–607

    2018

  12. [19]

    Iliescu and Z

    T. Iliescu and Z. Wang, Variational multiscale proper orthogonal decomposition: Navier-Stokes equations, Num. Meth. P.D.E.s 30 (2014), no. 2, 641–663

    2014

  13. [20]

    Ingimarson, L

    S. Ingimarson, L. G Rebholz, and T. Iliescu, Full and reduced order model consistency of the nonlinearity discretization in incompressible flows, Comput. Meth. Appl. Mech. Engrg. 401 (2022), 115620

    2022

  14. [21]

    Iollo, S

    A. Iollo, S. Lanteri, and J. A. D´ esid´ eri,Stability properties of POD–Galerkin approx- imations for the compressible Navier–Stokes equations , Theoret. Comput. Fluid Dyn. 13 (2000), no. 6, 377–396

    2000

  15. [22]

    John, Large eddy simulation of turbulent incompressible flows: analytical and nu- merical results for a class of LES models , Vol

    V. John, Large eddy simulation of turbulent incompressible flows: analytical and nu- merical results for a class of LES models , Vol. 34, Springer Science & Business Media, 2003

    2003

  16. [23]

    Kaneko, P.-H

    K. Kaneko, P.-H. Tsai, and P. Fischer, Towards model order reduction for fluid- thermal analysis , Nucl. Eng. Des. 370 (2020), 110866

    2020

  17. [24]

    verifiability

    M. Kaya and W. J. Layton, On “verifiability” of models of the motion of large eddies in turbulent flows , Differential and Integral Equations 15 (2002), no. 11, 1395 –1407. 39

    2002

  18. [26]

    B. Koc, C. Mou, H. Liu, Z. Wang, G. Rozza, and T. Iliescu, Verifiability of the data- driven variational multiscale reduced order model , J. Sci. Comput. 93 (2022), no. 2, 54

    2022

  19. [27]

    Kunisch and S

    K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (1999), no. 2, 345–371. MR1706822 (2000e:49042)

    1999

  20. [28]

    , Galerkin proper orthogonal decomposition methods for parabolic problems , Numer. Math. 90 (2001), no. 1, 117–148. MR1868765 (2003g:65118)

    2001

  21. [29]

    O. A. Ladyzhenskaya, The boundary value problems of mathematical physics , trans- lated by J. Lohwater, Applied Mathematical Sciences, vol. 49, Springer, New York, 1985

    1985

  22. [30]

    O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon & Breach (1969)

    1969

  23. [31]

    Layton, Introduction to the numerical analysis of incompressible viscous flows , Vol

    W. Layton, Introduction to the numerical analysis of incompressible viscous flows , Vol. 6, SIAM, 2008

    2008

  24. [32]

    K Lilly, The representation of small-scale turbulence in numerical simulation ex- periments, Proc

    D. K Lilly, The representation of small-scale turbulence in numerical simulation ex- periments, Proc. ibm sci. comput. symp. on environmental science, 1967, pp. 195– 210

    1967

  25. [33]

    J. L. Lions, Quelques m´ ethodes de r´ esolution des problemes aux limites non lin´ eaires (1969)

    1969

  26. [34]

    Manti, P.-H

    S. Manti, P.-H. Tsai, A. Lucantonio, and T. Iliescu, Symbolic regression of data- driven reduced order model closures for under-resolved, convection-dominated flows , arXiv preprint arXiv:2502.04703 (2025)

    2025 arXiv

  27. [35]

    J Minty, Monotone (nonlinear) operators in Hilbert space , Duke Math

    G. J Minty, Monotone (nonlinear) operators in Hilbert space , Duke Math. J. 29 (1962), no. 3, 341–346

    1962

  28. [36]

    C. Mou, E. Merzari, O. San, and T. Iliescu, An energy-based lengthscale for reduced order models of turbulent flows , Nucl. Eng. Des. 412 (2023), 112454

    2023

  29. [37]

    B. R. Noack, P. Papas, and P. A. Monkewitz, Low-dimensional Galerkin model of a laminar shear-layer , Technical Report 2002-01, ´Ecole Polytechnique F´ ed´ erale de Lausanne, 2002

    2002

  30. [38]

    E. J. Parish, M. Yano, I. Tezaur, and T. Iliescu, Residual-based stabilized reduced- order models of the transient convection-diffusion-reaction equation obtained through discrete and continuous projection , Arch. Comput. Methods Eng. (2024), 1–45

    2024

  31. [39]

    Prakash and Y

    A. Prakash and Y. J. Zhang, Projection-based reduced order modeling and data-driven artificial viscosity closures for incompressible fluid flows , Comput. Meth. Appl. Mech. Engrg. 425 (2024), 116930. 40

    2024

  32. [40]

    Protas, B

    B. Protas, B. R. Noack, and J. ¨Osth, Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows , J. Fluid Mech. 766 (2015), 337–367

    2015

  33. [41]

    Quaini, O

    A. Quaini, O. San, A. Veneziani, and T. Iliescu, Bridging large eddy simulation and reduced-order modeling of convection-dominated flows through spatial filtering: Review and perspectives, Fluids 9 (2024), no. 8

    2024

  34. [42]

    T. C. Rebollo and R. Lewandowski, Mathematical and numerical foundations of tur- bulence models and applications , Springer, 2014

    2014

  35. [43]

    T. C. Rebollo, E. D. ´Avila, and M. G. M´ armol, On a certified VMS-Smagorinsky reduced basis model with LPS pressure stabilisation, Appl. Numer. Math. 185 (2023), 365–385

    2023

  36. [44]

    T. C. Rebollo, E. D. Avila, M. G. M´ armol, F. Ballarin, and G. Rozza, On a certified Smagorinsky reduced basis turbulence model, SIAM J. Numer. Anal. 55 (2017), no. 6, 3047–3067

    2017

  37. [45]

    Reyes, P.-H

    J. Reyes, P.-H. Tsai, J. Novo, and T. Iliescu, A priori error bounds and parameter scalings for the time relaxation reduced order model , arXiv preprint arXiv:2411.08986 (2024)

    2024

  38. [46]

    Reyes and R

    R. Reyes and R. Codina, Projection-based reduced order models for flow problems: A variational multiscale approach , Comput. Meth. Appl. Mech. Engrg. 363 (2020), 112844

    2020

  39. [47]

    Sagaut, Large eddy simulation for incompressible flows: an introduction , Springer Science & Business Media, 2006

    P. Sagaut, Large eddy simulation for incompressible flows: an introduction , Springer Science & Business Media, 2006

    2006

  40. [48]

    Sanfilippo, I

    A. Sanfilippo, I. Moore, F. Ballarin, and T. Iliescu, Approximate deconvolution Leray reduced order model for convection-dominated flows , Finite Elem. Anal. Des. 226 (2023), 104021

    2023

  41. [49]

    Smagorinsky, General circulation experiments with the primitive equations: I

    J. Smagorinsky, General circulation experiments with the primitive equations: I. the basic experiment, Monthly weather review 91 (1963), no. 3, 99–164

    1963

  42. [50]

    Stabile, F

    G. Stabile, F. Ballarin, G. Zuccarino, and G. Rozza, A reduced order variational multiscale approach for turbulent flows , Adv. Comput. Math. 45 (2019), 2349–2368

    2019

  43. [51]

    Strazzullo, M

    M. Strazzullo, M. Girfoglio, F. Ballarin, T. Iliescu, and G. Rozza, Consistency of the full and reduced order models for evolve-filter-relax regularization of convection- dominated, marginally-resolved flows , Int. J. Num. Meth. Eng. 123 (2022), no. 14, 3148–3178

    2022

  44. [52]

    Temam, Navier-Stokes equations: Theory and numerical analysis, Vol

    R. Temam, Navier-Stokes equations: Theory and numerical analysis, Vol. 2, American Mathematical Society, 2001

    2001

  45. [53]

    P. H. Tsai, P. Fischer, and T. Iliescu, A time-relaxation reduced order model for the turbulent channel flow , J. Comput. Phys. 521 (2025), 113563

    2025

  46. [54]

    P. H. Tsai and P. Fischer, Parametric model-order-reduction development for unsteady convection, Front. Phys. 10 (2022), 903169. 41

    2022

  47. [55]

    P.-H. Tsai, P. Fischer, and E. Solomonik, Accelerating the Galerkin reduced- order model with the tensor decomposition for turbulent flows , arXiv preprint arXiv:2311.03694 (2023)

    2023 arXiv

  48. [56]

    Ullmann and J

    S. Ullmann and J. Lang, A POD-Galerkin reduced model with updated coefficients for Smagorinsky LES, V European conference on computational fluid dynamics, ECCO- MAS CFD, 2010, pp. 2010

    2010

  49. [57]

    Volkwein, Proper orthogonal decomposition: Theory and reduced-order modelling , Lecture Notes, University of Konstanz (2013)

    S. Volkwein, Proper orthogonal decomposition: Theory and reduced-order modelling , Lecture Notes, University of Konstanz (2013). http://www.math.uni-konstanz.de/ numerik/personen/volkwein/teaching/POD-Book.pdf

    2013

  50. [58]

    Z. Wang, I. Akhtar, J. Borggaard, and T. Iliescu, Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison , Comput. Meth. Appl. Mech. Engrg. 237 (2012), 10–26

    2012

  51. [59]

    X. Xie, D. Wells, Z. Wang, and T. Iliescu, Approximate deconvolution reduced order modeling, Comput. Methods Appl. Mech. Engrg. 313 (2017), 512–534

    2017

  52. [60]

    , Numerical analysis of the Leray reduced order model, J. Comput. Appl. Math. 328 (2018), 12–29. 42

    2018

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