Second order reduced model via incremental projection for Navier Stokes

Carlos N\'u\~nez Fern\'andez; Chuanju Xu; Mejdi Aza\"iez; Samuele Rubino; Yayu Guo

REVIEW 5 major objections 4 minor 19 references

Second order reduced model via incremental projection for Navier Stokes

T0 review · 5 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A POD-based reduced-order incremental projection scheme for the Stokes equations is stable, and its velocity and pressure errors are bounded—up to an explicit exponential factor—by the POD projection error whenever the reduced velocity and

desk verdict A plausible but sloppy ROM paper: the BDF2-incremental-projection extension is real, but the numerical validation and the printed reduced system do not hold up. read the letter →

arxiv 2512.10473 v1 pith:NKE36C4M submitted 2025-12-11 math.NA cs.NA

Mejdi Azaïez , Yayu Guo , Carlos Núñez Fernández , Samuele Rubino , Chuanju Xu This is my paper · ORCID

classification math.NAcs.NA MSC 65N3065M1576D07

keywords reducedordermodelproperorthogonaldecompositionincrementalprojectionStokesequationsBDF2errorestimatesaturationpropertyincompressibleflow

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 proposes a reduced-order model for incompressible flow that combines incremental projection time-splitting with proper orthogonal decomposition. It proves that the reduced scheme is stable and that, under a saturation condition on the reduced spaces, the velocity and pressure errors are controlled by the POD truncation error—the sum of the omitted POD eigenvalues. This means the reduced model's accuracy is essentially as good as the POD basis itself, and pressure is computed explicitly without needing an inf-sup condition. Numerical experiments confirm second-order time convergence and show that a few modes give very small errors.

What carries the argument

The central object is the saturation property (Definition 3.1): a pair of finite-dimensional subspaces (Y,Z) of a Hilbert space is saturated when ||y||+||z|| ≤ C||y+z|| for all y,z, equivalently |(y,z)| ≤ α||y||||z|| with α<1—an angle condition. The paper uses this with Y=Ur and Z=span{∇ψ_i} to control pressure–velocity cross terms. Around this, the method builds on an incremental (Goda-type) BDF2 projection scheme for the Stokes equations, H1-inner-product POD bases for both velocity and pressure, and a three-term recursion inequality plus Gronwall's lemma to handle the time-stepping.

What would settle it

Compute α = sup_{y∈Ur, z∈∇Qr} |(y,z)_X| / (||y||_X ||z||_X) for the H1-based POD spaces built from the paper's snapshot ensembles; if for typical snapshot sets one finds α ≥ 3/8, the theorem's hypothesis is violated and the stated error bound is not guaranteed. Additionally, with α in hand, one can compare the measured ROM errors against the right-hand side of (3.26) at fixed ν and τ to test the bound directly.

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Extended reading notes

Core claim

Theorem 3.2: if the saturation constant α of the reduced velocity space Ur and the pressure-gradient space span{∇ψ_1,...,∇ψ_r} is strictly less than 3/8, then the difference between the reduced-order solution and the full finite-element solution satisfies a bound where the summed velocity and pressure errors are at most C(α) e^{((7τ+4α)/2)T} times the sum of the omitted velocity and pressure POD eigenvalues, each weighted by norms of the corresponding POD modes. This makes the POD truncation error the sole driver of the ROM error, up to a stability factor.

Load-bearing premise

The load-bearing premise is that the reduced velocity space and the pressure-gradient space satisfy the saturation property with constant α < 3/8, and that the time step is below a threshold depending on α—conditions neither proven for POD bases nor checked in the numerical experiments, which instead use L2-based POD.

Editorial extensions

If this is right

The ROM error can be reduced predictably by adding POD modes; the estimate gives the per-mode contribution as weighted eigenvalues, so mode selection is a matter of truncating small eigenvalues.
No inf-sup/LBB condition is required in the reduced pair; pressure is obtained explicitly through a pressure-stiffness update, preserving the decoupling that makes projection methods efficient.
The scheme is second-order in time; the numerical tables show convergence rates near 1.99 for velocity and 1.98 for pressure.
Very few modes suffice on smooth problems: two velocity modes give relative errors around 10^-7, and about four pressure modes reach the same level.
The method extends to parametric settings: in the lid-driven cavity test, the ROM tracks the projection error when interpolating in Reynolds number and even when extrapolating in time.

Reading between the lines

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

The saturation constant α is computable from the POD bases; a practical implementation should measure it, because if α ≥ 3/8 the proven bound does not apply and the error may not follow the POD truncation.
The numerical experiments generate POD modes with the L2 inner product, while the theory assumes H1 (gradient) inner products; verifying the theory numerically would require repeating the tests with H1-based POD and checking α.
The exponential factor e^{((7τ+4α)/2)T} implies the error bound is only useful over moderate time horizons; for long-time integration the bound becomes very pessimistic, suggesting the need for time-domain decomposition or a sharper analysis.
The rigorous error analysis is for the linear Stokes problem; the lid-driven cavity test adds a nonlinear convection term via extrapolation, so a full Navier–Stokes error estimate remains open, as the authors themselves indicate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a legitimate extension of the authors' earlier Goda-based ROM to BDF2 incremental projection, with a stability proof and an error estimate worth reading. But as it stands, the numerical section is invalid, the reduced system as printed has a sign and dimension error, and the central error bound rests on an unverified saturation assumption. Don't take the numerics as evidence.

What's new: the ROM construction and error analysis for this specific time-splitting scheme. The stability argument (Theorem 3.1) follows the usual three-term recursion, and Theorem 3.2 gives the expected bound: ROM error controlled by POD projection error, up to an exponential factor. That is not a breakthrough, but it is a reasonable extension within a useful line of work.

Where it falls short. First, the manufactured solution (4.1) is not divergence-free. Both components are given by the same expression, and ∂x u + ∂y v is not zero. Since the scheme enforces incompressibility, the 'exact' solution cannot be approximated in the expected sense; the convergence table is therefore meaningless. Second, the printed reduced system (3.5a) has a wrong sign on the pressure term and a dimension mismatch: the matrix P is Np×Nu, so the term should be -1/3 P^T(7bn - 5bn-1 + bn-2), not +1/3 P(...). Someone trying to implement from (3.5) would get stuck. Third, the theory assumes H1-POD (inner products with gradients), while the experiments explicitly generate modes in the L2 norm. So the numerical tests do not exercise the spaces the analysis requires. Fourth, Theorem 3.2 is conditional on a saturation constant α < 3/8 for the pair (U_r, ∇Q_r); the paper neither proves this nor offers a way to check it. That is a serious gap in the main theorem. Also minor: initial pressure data for the ROM are not specified, and the lid-driven cavity test adds convection while the analysis covers only Stokes.

Who should read this: someone working on projection-based POD-ROMs who wants to see the BDF2-incremental variant analyzed; the proof technique may be reusable. But I would not cite the numerical results, and I would not recommend accepting the paper in this form. A serious referee should see it though—the core idea is meaningful and the defects are fixable.

Referee Report

5 major / 4 minor

Summary. The paper proposes a POD-based reduced-order model for the unsteady Stokes equations built on a BDF2 incremental projection scheme. It presents semi-discrete and fully discrete formulations, a stability estimate (Theorem 3.1), and an a priori error estimate (Theorem 3.2) in which the ROM error is bounded by POD truncation eigenvalues up to a factor depending on a saturation constant alpha < 3/8. Numerical experiments with a manufactured solution and a lid-driven cavity are reported. The main claim is that the reduced velocity and pressure are accurate with few modes and that the scheme preserves second-order accuracy in time.

Significance. If the main theorem were fully established, the paper would be a useful contribution: it combines a popular pressure-correction time-splitting scheme with POD, and the claimed bound by POD truncation error is an attractive and practically relevant statement. The paper also attempts a complete stability analysis. However, the numerical validation is invalid because the manufactured solution is not divergence-free, the printed reduced algebraic system is inconsistent with the Galerkin weak form, the experiments use L2-POD while the theory uses H1-POD, and the central error estimate rests on an unverified saturation hypothesis and contains gaps in the Gronwall argument. As it stands, the central claims are not supported.

major comments (5)

[Section 4.1, Eq. (4.1)] The manufactured solution is not divergence-free. Both components are printed as u=v=cos(t) pi sin^2(pi x) sin(2 pi y), whose divergence is 2 pi^2 cos(t) sin(pi x)(cos(pi x) sin(2 pi y)+sin(pi x) cos(2 pi y)), not identically zero. It therefore violates (2.1b) and is not a solution of the Stokes problem. Consequently the convergence rates in Table 1 and the ROM errors in Figures 2-3 do not measure errors against a valid exact solution. A divergence-free manufactured solution is required to validate the scheme or the theory.
[Section 3.2, Eqs. (3.4a)-(3.5b)] The reduced algebraic system is inconsistent with the stated weak form. The pressure term in (3.4a) is -1/3 (p, div v_r). With P_ij=(psi_i, div phi_j), this contributes -1/3 P^T b, not +1/3 P b as printed in (3.5a). Similarly, (div u^{n+1}, psi_i) with the stated definition of P gives P a^{n+1}, so (3.5b) should not contain P^T. As printed, the system does not represent the Galerkin equations (3.4). Any implementation following (3.5) solves a different ROM, so the numerical results do not validate the printed scheme.
[Sections 3.1 and 4.1.1/4.2.1] The theory and experiments use different POD inner products. Section 3.1 constructs POD bases with X=H^1-seminorm and Y=H^1-seminorm, with orthonormality (nabla phi_i, nabla phi_j)=delta_ij; but Sections 4.1.1 and 4.2.1 explicitly state that 'POD modes are generated in L2-norm for velocity and pressure'. The POD projection error estimates and the saturation property in Theorem 3.2 are formulated for the H1-POD spaces, so the experiments do not test the spaces appearing in the theory. The paper must either adapt the analysis to L2-POD or run the numerical tests with H1-POD.
[Definition 3.1, Lemma 3.1, Theorem 3.2] The saturation property is assumed, not verified, for the specific reduced spaces U_r and span{nabla psi_1,...,nabla psi_r}. Definition 3.1 requires Y cap Z = {0} and (3.23), and Theorem 3.2 assumes the saturation constant satisfies alpha < 3/8. The paper proves no such property for POD bases and gives no computational estimate of alpha. This is load-bearing: the absorption step leading to (3.46) requires positivity of 1-17 tau/4-alpha, and the Gronwall coefficient depends on alpha. Without a proof, a verifiable criterion, or an estimate of alpha for the actual POD spaces, the advertised error bound is conditional on an unestablished geometric hypothesis.
[Theorem 3.2 proof, Eqs. (3.41), (3.51), (3.52)] Two central steps in the proof are not justified. First, the estimate (3.41) for A6 is not a consequence of Lemma 3.1: A6 is an L2 pairing between an element of U_r and an element of span{nabla Q_r}, whereas Lemma 3.1 controls the X-inner product. If X=H^1, the controlled inner product is (nabla . , nabla .); if X=L^2, the POD construction is different. No inverse or norm-equivalence inequality is stated. Second, the discrete Gronwall step to (3.52) is inconsistent with (3.51): the coefficient on the energy-type sum is (7 tau + 4 alpha)/4, so discrete Gronwall gives a factor (1+(7 tau+4 alpha)/4)^{N_t} ~ exp((alpha+7 tau/4)T/tau), not exp((7 tau+4 alpha)T/2). The exponent in (3.52) has the wrong scaling in T and tau, which affects the validity of the stated error bound.

minor comments (4)

[Section 4.2] The lid-driven cavity test solves the nonlinear Navier-Stokes equations with the extrapolated convection term ((2 u^n - u^{n-1}) . grad) u^{n+1}, while the stability and error analyses in Section 3 are for the linear Stokes problem. These results should be labeled as heuristic and outside the scope of Theorem 3.2.
[Section 3.1] The summation indices in the POD basis formulas run from k=0 to N_t+1, but the snapshots are defined for indices 1 to N_t+1; u^0 is not defined. The index range should be corrected.
[Section 4.1.1] The statement that 81 snapshots are collected from storing every fourth FOM simulation in [0.21,1] appears inconsistent with tau=1e-2; the number of stored snapshots and the time interval should be reconciled.
[Abstract and Conclusion] The abstract claims numerical validation of second-order convergence in time, but Table 1 tests the full-order scheme, not the ROM; the ROM experiments show errors versus mode number rather than time-step convergence. The text should distinguish these.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central error-transfer theorem is conditional and self-contained; the main issues are an unverified saturation hypothesis and a self-citation that is not itself the argument.

full rationale

Theorem 3.2 is a conditional error-transfer result: it starts from the full-order scheme (2.6), the POD reduced scheme (3.4), and the POD projection errors (eigenvalue tails), and proves an energy inequality whose right-hand side is the projection error. Nothing is fitted from ROM output and then called a prediction; the constants C(α) are not calibrated against the reported errors. The saturation property (Definition 3.1, Eq. (3.23)) is assumed, not proved, for Y=U_r and Z=span{∇ψ1,...,∇ψr}, and the condition α<3/8 is used to keep coefficients positive (e.g., the absorbing step leading to (3.46) and the condition τ<4(1−8α/3)/17). This is an unverified hypothesis rather than a circular derivation; the theorem is honestly worded as an 'if' statement. Lemma 3.1 is cited to the authors' prior work [17], but it is a general equivalence between saturation and an angle bound, and its assumptions do not include the target error estimate, so it is real, if self-referential, evidence. The numerical section generates POD modes in the L2 norm (Sections 4.1.1 and 4.2.1) while the theory uses H1-seminorm inner products (Section 3.1), so the experiments do not test the saturation hypothesis; this is a validation gap, not circularity. Overall, no step reduces by construction to its own input, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central result rests on a saturation assumption for the ROM spaces, the known accuracy of the full-order projection scheme, standard POD projection theory, and an incomplete specification of initial pressure data. No new entities or fitted constants are introduced.

assumptions (5)

domain assumption The reduced spaces U_r and Q_r satisfy the saturation property with constant α < 3/8 (Definition 3.1, Eq. 3.23).
Hypothesis of Theorem 3.2; the paper does not prove that POD bases satisfy it. It is used to control pressure–velocity cross terms (A6 and related bounds in the proof).
domain assumption The full-order BDF2 incremental projection scheme (2.6) is stable and second-order accurate in time (O(δt^2 + h^{l+1})), as established in [12].
The ROM error is measured against the full-order solution; the paper cites [12] for the full-order accuracy instead of proving it.
standard math POD projection error equals the sum of neglected eigenvalues: (1/(N_t+1))∑∥ũ^i - Π_v ũ^i∥^2_X = ∑_{j>N_u} λ_j, and analogously for pressure (Section 3.1).
Standard POD optimality result (Kunisch-Volkwein [13]); used to convert the ROM error bound into eigenvalue tails.
standard math Lemma 3.1 (saturation property equivalent to angle bound, |(y,z)_X| ≤ α∥y∥_X∥z∥_X) holds as stated in [17, Lemma 5.3].
Borrowed from the same group's prior work; not proved in the paper. Required for the error analysis.
domain assumption Initial pressure data p^0_h and p^{-1}_h exist and are available for the BDF2 recurrence (2.6) at n=1.
The scheme involves p^{n-2} at n=1; the paper only states that ũ^1, ũ^2, p^1, p^2 are obtained by a first-order scheme, leaving p^0 and p^{-1} unspecified.

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Cite this review

Pith. "Pith review of Second order reduced model via incremental projection for Navier Stokes." pith.science (2026). https://pith.science/paper/NKE36C4M

@misc{pith2026251210473,
  author       = {Pith},
  title        = {Pith review of: Second order reduced model via incremental projection for Navier Stokes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKE36C4M}},
  note         = {Machine review of arXiv:2512.10473}
}

read the original abstract

The numerical simulation of incompressible flows is challenging due to the tight coupling of velocity and pressure. Projection methods offer an effective solution by decoupling these variables, making them suitable for large-scale computations. This work focuses on reduced-order modeling using incremental projection schemes for the Stokes equations. We present both semi-discrete and fully discrete formulations, employing BDF2 in time and finite elements in space. A proper orthogonal decomposition (POD) approach is adopted to construct a reduced-order model for the Stokes problem. The method enables explicit computation of reduced velocity and pressure while preserving accuracy. We provide a detailed stability analysis and derive error estimates, showing second-order convergence in time. Numerical experiments are conducted to validate the theoretical results and demonstrate computational efficiency.

Figures

Figures reproduced from arXiv: 2512.10473 by the authors.

**Figure 1.** Decay of the normalized POD eigenvalues (left) and captured energy (right). comparison purposes, we also plot the l 2 (L 2 ) relative projection errors for both velocity and pressure in the same figure. It is worth noting that this error represents the best achievable error. 1 1.5 2 2.5 3 r 10-8 10-7 10-6 10-5 10-4 10-3 Relative Errors Velocity l2l2 velocity l2l2 proj vel 1 1.5 2 2.5 3 3.5 4 4.5 5 r 10-8 10-6 10-4 1… view at source ↗

**Figure 2.** l2l2 errors of velocity and pressure. We can also observe in Figure (2) that for the velocity case, using 2 basis functions results in negligible relative errors on the order of 10−7 . However, due to the slower decay of pressure eigenvalues with respect to the velocity ones, as we can see in Figure (1), to achieve the same order of error for the pressure, 4 basis functions are required. Finally, [PITH_FULL_IMAGE:f… view at source ↗

**Figure 3.** Comparative between the theoretical error estimator (3.26) and the velocity and pressure error depending on the number of modes (r). 4.2. Lid-driven cavity flow. The lid-driven cavity problem is a classical benchmark in computational fluid dynamics, widely studied for its rich flow structures and well-defined boundary conditions. This problem involves a square or rectangular domain where the fluid is driven by the … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Decay of the normalized POD eigenvalues (left) and captured energy (right). 4.2.2. Numerical Results. Once the POD modes are generated, the ROM is constructed by using the same time discretization as the FOM. We have evaluated the ROM in the same time interval as its c…

**Figure 5.** Figure 5: l2l2 errors of velocity and pressure. 2 2.5 3 3.5 4 10-8 10-7 10-6 10-5 10-4 10-3 10-2 l2 extrapolation error Velocity Re = 2000 Re = 4000 2 2.5 3 3.5 4 10-7 10-6 10-5 10-4 10-3 10-2 l2 extrapolation error Pressure Re = 2000 Re = 4000 [PITH_FULL_IMAGE:figures/full_fig…

**Figure 6.** Figure 6: l2 error extrapolation time Figure (7) show the l 2 (L 2 ) relative error of velocity and pressure computing in the same time interval as snapshots, [2, 3). In Figure (7), We can see that the maximum error for both velocity and pressure takes at the mean value between …

**Figure 7.** Figure 7: l2l2 relative error parameter extrapolation sufficient to attain high accuracy; for instance, negligible relative errors (∼ 10−7 ) were achieved using only two velocity modes. The classical lid-driven cavity problem served as a further benchmark to evaluate the ROM und…

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Aza ¨ ıez, T

M. Aza ¨ ıez, T. Chac´ on Rebollo, C. N´ u˜ nez Fern´ andez, and S. Rubino. A pod-based reduced order method applied to goda time-splitting scheme. Journal of Scientiﬁc Computing , 104(1):9, 2025

2025

[2] [2]

Fran¸ cois E. Cellier. Continuous system modeling . New York etc.: Springer-Verlag, 1991

1991

[3] [3]

A.J. Chorin. Numerical solution of the navier–stokes eq uations. Math. Comput , 22(4):745–762, 1968

1968

[4] [4]

John, and Julia Novo

Bosco Garc ´ ıa-Archilla, V. John, and Julia Novo. Error analysis of BDF schemes for the evolutionary incompressible Navier–Stokes equations. Preprint, arXiv:2506.16917 [ma th.NA] (2025), 2025

arXiv 2025

[5] [5]

Se cond order error bounds for POD-ROM methods based on ﬁrst order divided diﬀerences

Bosco Garc ´ ıa-Archilla, Volker John, and Julia Novo. Se cond order error bounds for POD-ROM methods based on ﬁrst order divided diﬀerences. Appl. Math. Lett. , 146:7, 2023. Id/No 108836

2023

[6] [6]

Glowinski

R. Glowinski. Finite element methods for incompressible viscous ﬂow . Elsevier Science B.V., Amsterdam, 2003

2003

[7] [7]

A multistep technique with implicit diﬀe rence schemes for calculating two- or three-dimensional cavity ﬂows

Katuhiko Goda. A multistep technique with implicit diﬀe rence schemes for calculating two- or three-dimensional cavity ﬂows. J. Comput. Phys. , 30:76–95, 1979

1979

[8] [8]

Error analysis of a frac tional time-stepping technique for incompressible ﬂows with variable density

J-L Guermond and Abner J Salgado. Error analysis of a frac tional time-stepping technique for incompressible ﬂows with variable density. SIAM Journal on Numerical Analysis , 49(3):917–944, 2011

2011