pith. machine review for the scientific record. sign in

arxiv: 2512.04592 · v2 · submitted 2025-12-04 · 🧮 math.NA · cs.NA· physics.comp-ph· physics.flu-dyn

Recognition: 2 theorem links

· Lean Theorem

Stable self-adaptive timestepping for Reduced Order Models for incompressible flows

Josep Plana-Riu , Henrik Rosenberger , Benjamin Sanderse , F.Xavier Trias
Authors on Pith no claims yet

Pith reviewed 2026-05-17 02:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-phphysics.flu-dyn
keywords reduced-order modelsadaptive timesteppingincompressible Navier-Stokesstability analysisprojection methodseigenvalue boundsself-adaptive methods
0
0 comments X

The pith

Projection-based reduced-order models of incompressible flows admit stable timesteps at least as large as their full-order models under linearization.

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

This paper presents RedEigCD, a self-adaptive timestepping method designed for reduced-order models of the incompressible Navier-Stokes equations. The approach adjusts the timestep by bounding the stability function of the integrator with spectral data from the reduced convective and diffusive operators, which keeps all computations at the small reduced scale. A central theoretical result proves, via the theorems of Bendixson and Rao, that under linearized assumptions the largest stable timestep for such projection-based ROMs is never smaller than the corresponding bound for the full-order model. Experiments on periodic and non-homogeneous cases confirm that the method can raise the timestep by factors up to 40 while accuracy remains intact.

Core claim

Under linearized assumptions the maximum stable timestep for projection-based ROMs is shown to be larger than or equal to that of their corresponding full-order models, proven by combining the theorems of Bendixson and Rao; this fact is exploited by RedEigCD, which adapts the timestep directly from the eigenbounds of the reduced operators and thereby preserves online efficiency.

What carries the argument

RedEigCD, which adapts the timestep by bounding the stability function of the time integrator using exact eigenbounds of the convective and diffusive reduced operators.

If this is right

  • Stable timestep increases up to a factor of 40 compared to the FOM are achieved without loss of accuracy.
  • All spectral computations remain at the reduced scale, so online efficiency is fully preserved.
  • The approach works for both periodic and non-homogeneous boundary conditions.
  • A direct link is established between linear stability theory and reduced-order modeling that enables systematic self-regulating integration.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same eigenvalue-bounding idea could be tested on non-projection ROMs or on other time integrators beyond those examined here.
  • If the linearization assumption is relaxed in future work, the observed factor-of-40 gains might shrink or require additional safeguards.
  • Engineering codes that already use ROMs for incompressible flow could adopt this method to cut wall-clock time by roughly the same factor shown in the experiments.

Load-bearing premise

The stability proof and timestep bound rely on linearized assumptions for the ROM operators.

What would settle it

A direct numerical comparison, under the stated linearization, in which a projection-based ROM becomes unstable at a timestep that the corresponding FOM can still take.

Figures

Figures reproduced from arXiv: 2512.04592 by Benjamin Sanderse, F.Xavier Trias, Henrik Rosenberger, Josep Plana-Riu.

Figure 1
Figure 1. Figure 1: Schematic illustration of Bendixson’s rectangle (gray) [6] for the eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphical representation of the range of reduced eigenvalues for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphical representation of the range of reduced singular values for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Singular values for the Re=1000 shear-layer roll-up up to M = 200. Moreover, note that even for the largest number of modes considered, M = 200, where the singular values have decayed up to O(10−9 ), the number of degrees of freedom is notably reduced compared to those of the FOM, which is 3 × 104 , considering both x− and y− velocity components as well as the pressure. The simulation has been run for Case… view at source ↗
Figure 5
Figure 5. Figure 5: Timestep ratio between Case B and Case A up to t = 20 (left) and evolution of the imaginary eigenbound (right) for the roll-up of a shear-layer at Re = 1000. theoretically derived in Section 3.2 holds: the eigenbound of the reduced convective terms steadily increases with an increase in the number of modes. For M = 200, however, in some points the imaginary eigenbound for the ROM is slightly larger than fo… view at source ↗
Figure 6
Figure 6. Figure 6: Timestep ratio between Case D and Case A for up to t = 20 for the roll-up of a shear-layer of Re = 1000 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Error comparison of the ROM results for M = 16 (left) and M = 200 (right) in the shear-layer roll-up of Re = 1000 for Case B, C, and D. In terms of efficiency, the ROM with M = 64 reduced the number of unknowns from 3 × 104 to 64, while simul￾taneously allowing timesteps up to almost 5 times larger than the FOM. This reduced the total number of timesteps, while the wall-clock time for each timestep was als… view at source ↗
Figure 8
Figure 8. Figure 8: Singular values for the numerical simulation of an actuator disk of Re [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Timestep ratio between Case B and Case A up to t = 100 (left) and evolution of the imaginary eigenbound (right) in the numerical simulation of a wind field of Re = 100. The magnitude of the ROM error, calculated with Eq.(40), is not affected by the variable ∆t compared to the reference result from [8], where ∆t was set to 4π/200. As shown in [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Error comparison of the ROM results for M = 16 (left) and M = 200 (right) in an actuator disk simulation with Re = 100 for Case B, Case C, and Case D. 16 32 64 128 200 M 0 25 50 75 100 125 150 175 λ λaiCr,i RedEigCD gershgorin 16 32 64 128 200 M 0 10 20 30 40 λ [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the actual eigenvalues of the reduced convective operator at [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
read the original abstract

This work introduces RedEigCD, the first self-adaptive timestepping technique specifically tailored for reduced-order models (ROMs) of the incompressible Navier-Stokes equations. Building upon linear stability concepts, the method adapts the timestep by directly bounding the stability function of the employed time integration scheme using exact spectral information of matrices related to the reduced operators. Unlike traditional error-based adaptive methods, RedEigCD relies on the eigenbounds of the convective and diffusive ROM operators, whose computation is feasible at reduced scale and fully preserves the online efficiency of the ROM. A central theoretical contribution of this work is the proof, based on the combined theorems of Bendixson and Rao, that, under linearized assumptions, the maximum stable timestep for projection-based ROMs is shown to be larger than or equal to that of their corresponding full-order models (FOMs). Numerical experiments for both periodic and non-homogeneous boundary conditions demonstrate that RedEigCD yields stable timestep increases up to a factor 40 compared to the FOM, without compromising accuracy. The methodology thus establishes a new link between linear stability theory and reduced-order modeling, offering a systematic path towards efficient, self-regulating ROM integration in incompressible flow simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces RedEigCD, a self-adaptive timestepping technique for projection-based reduced-order models (ROMs) of the incompressible Navier-Stokes equations. The method uses exact spectral information from the reduced convective and diffusive operators to bound the stability function of the time integrator, thereby adapting the timestep while preserving online efficiency. A central theoretical contribution is the proof, based on the combined Bendixson and Rao theorems under linearized assumptions, that the maximum stable timestep for such ROMs is at least as large as for the corresponding full-order models (FOMs). Numerical experiments on cases with periodic and non-homogeneous boundary conditions report stable timestep increases of up to a factor of 40 relative to the FOM without loss of accuracy.

Significance. If the central claim holds, the work establishes a direct link between linear stability theory and reduced-order modeling, providing a systematic, non-error-based approach to stable and efficient ROM time integration for incompressible flows. The method's reliance on reduced-scale eigenbounds is a strength that maintains the computational advantages of ROMs. The reported timestep gains of up to 40x, if reproducible, would represent a substantial practical advance for long-time simulations.

major comments (1)
  1. Abstract and theoretical section: The proof that the ROM maximum stable timestep is >= the FOM timestep under linearized assumptions invokes the Bendixson theorem (bounding real parts via the symmetric part) and Rao theorem (for imaginary parts) to conclude that the stability-region-limited timestep for the ROM is no stricter than for the FOM. However, this requires that the numerical range (or eigenvalue region) of the reduced non-normal operator lies inside a region whose stability limit is at least as permissive as the FOM's. Projection-based ROMs for incompressible flows routinely produce non-normal matrices whose numerical range can extend outside the convex hull of the FOM eigenvalues, particularly with truncated POD bases or non-homogeneous boundary conditions. The manuscript should explicitly verify or prove that the required inclusion holds for the projected operators, or supply
minor comments (2)
  1. The abstract states that the method 'fully preserves the online efficiency of the ROM'; a brief complexity count or timing table in the numerical section would make this concrete.
  2. Clarify in the main text (beyond the abstract) the precise linearized assumptions on the convective and diffusive ROM operators and how they are enforced in the experiments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The major comment raises an important point about the numerical range of non-normal reduced operators, and we address it directly below with clarifications and planned revisions.

read point-by-point responses

  1. Referee: Abstract and theoretical section: The proof that the ROM maximum stable timestep is >= the FOM timestep under linearized assumptions invokes the Bendixson theorem (bounding real parts via the symmetric part) and Rao theorem (for imaginary parts) to conclude that the stability-region-limited timestep for the ROM is no stricter than for the FOM. However, this requires that the numerical range (or eigenvalue region) of the reduced non-normal operator lies inside a region whose stability limit is at least as permissive as the FOM's. Projection-based ROMs for incompressible flows routinely produce non-normal matrices whose numerical range can extend outside the convex hull of the FOM eigenvalues, particularly with truncated POD bases or non-homogeneous boundary conditions. The manuscript should explicitly verify or prove that the required inclusion holds for the projected operators, or supply

    Authors: We thank the referee for identifying this subtlety. Our proof applies the Bendixson and Rao theorems directly to the symmetric and skew-symmetric parts of the reduced convective and diffusive operators, yielding explicit bounds on their numerical ranges without assuming containment in the FOM numerical range a priori. The resulting stability timestep bound for the ROM is therefore derived independently and shown to be at least as permissive. While general projections can enlarge the numerical range, the specific structure arising from the divergence-free POD basis and the linearized incompressible operator ensures the relevant stability-limiting quantities (maximum real part and imaginary extent) do not exceed those of the FOM under the stated assumptions. To strengthen the presentation we will add a short remark in Section 3 clarifying this operator-specific application and include a supplementary numerical check of numerical-range boundaries for the reported test cases. revision: partial

Circularity Check

0 steps flagged

No circularity: ROM timestep bound derived from external Bendixson-Rao theorems on linearized operators

full rationale

The paper's central claim is a proof, under linearized assumptions, that the maximum stable timestep for projection-based ROMs is at least as large as for the corresponding FOMs. This is obtained by applying the existing Bendixson and Rao theorems to bound the real and imaginary parts of eigenvalues of the reduced convective and diffusive operators, then using those bounds to constrain the stability function of the time integrator. The derivation invokes no self-citation for the uniqueness or validity of the bound, performs no fitting of parameters that are later renamed as predictions, and does not define the target quantity in terms of itself. The RedEigCD adaptation itself computes eigenbounds directly from the reduced matrices at online cost, preserving independence from the FOM stability limit. The overall chain is therefore self-contained and relies on external mathematical results rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on linearized assumptions for stability analysis and the applicability of Bendixson and Rao theorems to the reduced operators; no free parameters or new physical entities are mentioned in the abstract.

axioms (1)
  • domain assumption Linearized assumptions hold for the stability analysis of the projection-based ROMs
    The proof that ROM maximum stable timestep is at least as large as the FOM one is stated to rely on these assumptions.

pith-pipeline@v0.9.0 · 5530 in / 1304 out tokens · 41161 ms · 2026-05-17T02:03:05.367434+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 1 internal anchor

  1. [1]

    Hairer, G

    E. Hairer, G. Wanner, S. P. Noersett, Solving Ordinary Differential Equations I, volume 8 ofSpringer Series in Computational Math- ematics, Springer Berlin Heidelberg, Berlin, Heidelberg, 1993. URL:http://link.springer.com/10.1007/978-3-540-78862-1. doi:10.1007/978-3-540-78862-1

    work page doi:10.1007/978-3-540-78862-1 1993

  2. [2]

    J. C. Butcher, Numerical methods for ordinary differential equations, 3rd ed ed., Wiley, Chichester, West Sussex, 2016

    work page 2016

  3. [3]

    Dormand, P

    J. Dormand, P. Prince, A family of embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics 6 (1980) 19–26. URL:https://linkinghub.elsevier.com/retrieve/pii/0771050X80900133. doi:10.1016/0771-050X(80)90013-3

    work page doi:10.1016/0771-050x(80)90013-3 1980

  4. [4]

    Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing 4 (1969) 93–106

    E. Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing 4 (1969) 93–106. URL:http://link.springer.com/10.1007/BF02234758. doi:10.1007/BF02234758

    work page doi:10.1007/bf02234758 1969

  5. [5]

    Courant, K

    R. Courant, K. Friedrichs, H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen 100 (1928) 32–74. doi:10.1007/BF01448839

    work page doi:10.1007/bf01448839 1928

  6. [6]

    F. X. Trias, O. Lehmkuhl, A Self-Adaptive Strategy for the Time Integration of Navier-Stokes Equations, Numerical Heat Transfer, Part B: Fundamentals 60 (2011) 116–134. URL:http://www.tandfonline.com/doi/abs/10.1080/10407790.2011.594398. doi:10.1080/ 10407790.2011.594398

    work page doi:10.1080/10407790.2011.594398 2011

  7. [7]

    F. X. Trias, X. Álvarez Farré, A. Alsalti-Baldellou, A. Gorobets, A. Oliva, An efficient eigenvalue bounding method: CFL condition revisited, Computer Physics Communications 305 (2024). doi:10.1016/j.cpc.2024.109351, publisher: Elsevier B.V

    work page doi:10.1016/j.cpc.2024.109351 2024

  8. [8]

    Sanderse, Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods, Journal of Computational Physics 421 (2020)

    B. Sanderse, Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods, Journal of Computational Physics 421 (2020). doi:10.1016/j.jcp.2020.109736, arXiv: 1909.11462 Publisher: Academic Press Inc. 21

    work page doi:10.1016/j.jcp.2020.109736 2020

  9. [9]

    Leblond, C

    C. Leblond, C. Allery, C. Inard, An optimal projection method for the reduced-order modeling of incompressible flows, Computer Meth- ods in Applied Mechanics and Engineering 200 (2011) 2507–2527. URL:https://www.sciencedirect.com/science/article/pii/ S0045782511001617. doi:10.1016/j.cma.2011.04.020

    work page doi:10.1016/j.cma.2011.04.020 2011

  10. [10]

    Reyes, R

    R. Reyes, R. Codina, Projection-based reduced order models for flow problems: A variational multiscale approach, Computer Meth- ods in Applied Mechanics and Engineering 363 (2020) 112844. URL:https://www.sciencedirect.com/science/article/pii/ S0045782520300256. doi:10.1016/j.cma.2020.112844

    work page doi:10.1016/j.cma.2020.112844 2020

  11. [11]

    Prakash, Y

    A. Prakash, Y . J. Zhang, Projection-based reduced order modeling and data-driven artificial viscosity closures for incompressible fluid flows, Computer Methods in Applied Mechanics and Engineering 425 (2024) 116930. URL:https://www.sciencedirect.com/science/ article/pii/S0045782524001865. doi:10.1016/j.cma.2024.116930

    work page doi:10.1016/j.cma.2024.116930 2024

  12. [12]

    J. L. Lumley, The structure of inhomogeneous turbulence, in: Atmospheric Turbulence and Wave Propagation, Moscow, 1967, pp. 166–178

    work page 1967

  13. [13]

    Sirovich, Turbulence and the dynamics of coherent structures

    L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Quarterly of Applied Mathematics 45 (1987) 561–571. URL:https://www.ams.org/qam/1987-45-03/S0033-569X-1987-0910462-6/. doi:10.1090/qam/910462

    work page doi:10.1090/qam/910462 1987

  14. [14]

    Holmes, J

    P. Holmes, J. L. Lumley, G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1 ed., Cambridge Uni- versity Press, 1996. URL:https://www.cambridge.org/core/product/identifier/9780511622700/type/book. doi:10.1017/ CBO9780511622700

    work page arXiv 1996

  15. [15]

    Quarteroni, G

    A. Quarteroni, G. Rozza, Numerical solution of parametrized Navier–Stokes equations by reduced basis methods, Numerical Methods for Partial Differential Equations 23 (2007) 923–948. URL:https://onlinelibrary.wiley.com/doi/10.1002/num.20249. doi:10. 1002/num.20249

    work page doi:10.1002/num.20249 2007

  16. [16]

    A. Manzoni, An efficient computational framework for reduced basis approximation anda posteriorierror estimation of parametrized Navier–Stokes flows, ESAIM: Mathematical Modelling and Numerical Analysis 48 (2014) 1199–1226. URL:http://www.esaim-m2an. org/10.1051/m2an/2014013. doi:10.1051/m2an/2014013

    work page doi:10.1051/m2an/2014013 2014

  17. [17]

    Lassila, A

    T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza, Model Order Reduction in Fluid Dynamics: Challenges and Perspectives, in: Reduced Order Methods for Modeling and Computational Reduction, Springer International Publishing, Cham, 2014, pp. 235–273. URL:http: //link.springer.com/10.1007/978-3-319-02090-7. doi:10.1007/978-3-319-02090-7

    work page doi:10.1007/978-3-319-02090-7 2014

  18. [18]

    Amsallem, C

    D. Amsallem, C. Farhat, Stabilization of projection-based reduced-order models, International Journal for Numerical Methods in Engineer- ing 91 (2012) 358–377. URL:https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4274. doi:10.1002/nme.4274, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.4274

    work page doi:10.1002/nme.4274 2012

  19. [19]

    Rezaian, M

    E. Rezaian, M. Wei, A global eigenvalue reassignment method for the stabilization of nonlinear reduced-order models, International Journal for Numerical Methods in Engineering 122 (2021) 2393–2416. URL:https://onlinelibrary.wiley.com/doi/abs/10.1002/nme

    work page doi:10.1002/nme 2021

  20. [20]

    doi:10.1002/nme.6625, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.6625

    work page doi:10.1002/nme.6625

  21. [21]

    Z. Wang, I. Akhtar, J. Borggaard, T. Iliescu, Proper orthogonal decomposition closure models for turbulent flows: A numerical compar- ison, Computer Methods in Applied Mechanics and Engineering 237-240 (2012) 10–26. URL:https://linkinghub.elsevier.com/ retrieve/pii/S0045782512001429. doi:10.1016/j.cma.2012.04.015

    work page doi:10.1016/j.cma.2012.04.015 2012

  22. [22]

    S. E. Ahmed, S. Pawar, O. San, A. Rasheed, T. Iliescu, B. R. Noack, On closures for reduced order models—A spectrum of first-principle to machine-learned avenues, Physics of Fluids 33 (2021) 091301. URL:https://pubs.aip.org/pof/article/33/9/091301/1030870/ On-closures-for-reduced-order-models-A-spectrum-of. doi:10.1063/5.0061577

    work page doi:10.1063/5.0061577 2021

  23. [23]

    Rosenberger, B

    H. Rosenberger, B. Sanderse, No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions, Journal of Computational Physics 491 (2023) 112405. URL:https://linkinghub.elsevier.com/retrieve/ pii/S0021999123005004. doi:10.1016/j.jcp.2023.112405

    work page doi:10.1016/j.jcp.2023.112405 2023

  24. [24]

    R. B. Klein, B. Sanderse, Energy-conserving hyper-reduction and temporal localization for reduced order models of the incompressible Navier-Stokes equations, Journal of Computational Physics 499 (2024) 112697. URL:https://www.sciencedirect.com/science/ article/pii/S0021999123007921. doi:10.1016/j.jcp.2023.112697

    work page doi:10.1016/j.jcp.2023.112697 2024

  25. [25]

    Farhat, T

    C. Farhat, T. Chapman, P. Avery, Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, International Journal for Numerical Methods in Engineering 102 (2015) 1077–1110. URL:https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4820. d...

    work page doi:10.1002/nme.4820 2015

  26. [26]

    Barrault, Y

    M. Barrault, Y . Maday, N. C. Nguyen, A. T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretiza- tion of partial differential equations, Comptes Rendus Mathematique 339 (2004) 667–672. URL:https://www.sciencedirect.com/ science/article/pii/S1631073X04004248. doi:10.1016/j.crma.2004.08.006

    work page doi:10.1016/j.crma.2004.08.006 2004

  27. [27]

    Chaturantabut, D

    S. Chaturantabut, D. C. Sorensen, A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction, SIAM Journal on Numerical Analysis 50 (2012) 46–63. URL:https://epubs.siam.org/doi/10.1137/110822724. doi:10.1137/110822724, publisher: Society for Industrial and Applied Mathematics

    work page doi:10.1137/110822724 2012

  28. [28]

    Ahmed, O

    M. Ahmed, O. San, Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks, Computational and Applied Mathematics 37 (2018) 6870–6902. URL:http://link.springer.com/10.1007/ s40314-018-0718-z. doi:10.1007/s40314-018-0718-z

    work page doi:10.1007/s40314-018-0718-z 2018

  29. [29]

    Stabile, F

    G. Stabile, F. Ballarin, G. Zuccarino, G. Rozza, A reduced order variational multiscale approach for turbulent flows, Advances in Com- putational Mathematics 45 (2019) 2349–2368. URL:http://link.springer.com/10.1007/s10444-019-09712-x. doi:10.1007/ s10444-019-09712-x

    work page doi:10.1007/s10444-019-09712-x 2019

  30. [30]

    X. Li, Y . Xu, M. Feng, A Pressure-Stabilized Continuous Data Assimilation Reduced Order Model for Incompressible Navier–Stokes Equations, Journal of Scientific Computing 103 (2025) 10. URL:https://link.springer.com/10.1007/s10915-025-02828-x. doi:10.1007/s10915-025-02828-x

    work page doi:10.1007/s10915-025-02828-x 2025

  31. [31]

    C. Bach, L. Song, T. Erhart, F. Duddeck, Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models, 2018. URL:https://arxiv.org/abs/1806.11404. doi:10.48550/ARXIV. 1806.11404, version Number: 1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2018

  32. [32]

    Stabile, G

    G. Stabile, G. Rozza, Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equa- 22 tions, Computers & Fluids 173 (2018) 273–284. URL:https://linkinghub.elsevier.com/retrieve/pii/S0045793018300422. doi:10.1016/j.compfluid.2018.01.035

    work page doi:10.1016/j.compfluid.2018.01.035 2018

  33. [33]

    Bendixson, Sur les racines d’une équation fondamentale, Acta Mathematica 25 (1902) 359–365

    I. Bendixson, Sur les racines d’une équation fondamentale, Acta Mathematica 25 (1902) 359–365. URL:http://projecteuclid.org/ euclid.acta/1485882119. doi:10.1007/BF02419030

    work page doi:10.1007/bf02419030 1902

  34. [34]

    Rao, Separation theorems for singular values of matrices and their applications in multivariate analysis, Journal of Multivariate Analysis 9 (1979) 362–377

    C. Rao, Separation theorems for singular values of matrices and their applications in multivariate analysis, Journal of Multivariate Analysis 9 (1979) 362–377. URL:https://linkinghub.elsevier.com/retrieve/pii/0047259X79900940. doi:10.1016/0047-259X(79) 90094-0

    work page doi:10.1016/0047-259x(79 1979

  35. [35]

    F. X. Trias, O. Lehmkuhl, A. Oliva, C. D. Pérez-Segarra, R. W. Verstappen, Symmetry-preserving discretization of Navier-Stokes equations on collocated unstructured grids, Journal of Computational Physics 258 (2014) 246–267. doi:10.1016/j.jcp.2013.10.031, publisher: Academic Press Inc

    work page doi:10.1016/j.jcp.2013.10.031 2014

  36. [36]

    R. W. C. P. Verstappen, A. E. P. Veldman, Symmetry-preserving discretization of turbulent flow, Journal of Computational Physics 187 (2003) 343–368. doi:10.1016/S0021-9991(03)00126-8, publisher: Academic Press Inc

    work page doi:10.1016/s0021-9991(03)00126-8 2003

  37. [37]

    A. J. Chorin, A numerical method for solving incompressible viscous flow problems, Journal of Computational Physics 2 (1967) 12–26. URL: https://linkinghub.elsevier.com/retrieve/pii/002199916790037X. doi:10.1016/0021-9991(67)90037-x, publisher: Else- vier BV

    work page doi:10.1016/0021-9991(67)90037-x 1967

  38. [38]

    Sanderse, B

    B. Sanderse, B. Koren, Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations, Journal of Computational Physics 231 (2012) 3041–3063. doi:10.1016/j.jcp.2011.11.028, publisher: Academic Press Inc

    work page doi:10.1016/j.jcp.2011.11.028 2012

  39. [39]

    S. D. Agdestein, B. Sanderse, Discretize first, filter next: Learning divergence-consistent closure models for large-eddy simulation, Journal of Computational Physics 522 (2025) 113577. URL:https://www.sciencedirect.com/science/article/pii/S0021999124008258. doi:10.1016/j.jcp.2024.113577

    work page doi:10.1016/j.jcp.2024.113577 2025

  40. [40]

    Hairer, G

    E. Hairer, G. Wanner, Solving Ordinary Differential Equations II, volume 14 ofSpringer Series in Computational Mathematics, Springer, Berlin, Heidelberg, 1996. URL:http://link.springer.com/10.1007/978-3-642-05221-7. doi:10.1007/978-3-642-05221-7

    work page doi:10.1007/978-3-642-05221-7 1996

  41. [41]

    J. F. B. M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT 31 (1991) 482–528

    work page 1991

  42. [42]

    Gerschgorin, Über die Abgrenzung fer Eigenwerte einer Matrix, Bulletin de l’Académie des Sciences de l’URSS: Classe des sciences mathématiques et na (1931) 749–754

    S. Gerschgorin, Über die Abgrenzung fer Eigenwerte einer Matrix, Bulletin de l’Académie des Sciences de l’URSS: Classe des sciences mathématiques et na (1931) 749–754

    work page 1931

  43. [43]

    Benner, S

    P. Benner, S. Gugercin, K. Willcox, A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems, SIAM Review 57 (2015) 483–531. URL:http://epubs.siam.org/doi/10.1137/130932715. doi:10.1137/130932715

    work page doi:10.1137/130932715 2015

  44. [44]

    Kunisch, S

    K. Kunisch, S. V olkwein, Optimal snapshot location for computing POD basis functions, ESAIM: Mathematical Modelling and Numerical Analysis 44 (2010) 509–529. URL:http://www.esaim-m2an.org/10.1051/m2an/2010011. doi:10.1051/m2an/2010011

    work page doi:10.1051/m2an/2010011 2010

  45. [45]

    H. Rosenberger, Advances in Structure-Preserving Reduced Order Models for the Incompressible Navier-Stokes Equations, Master’s thesis, Technische Universiteit Eindhoven, Eindhoven, 2021. URL:https://research.tue.nl/en/studentTheses/ advances-in-structure-preserving-reduced-order-models-for-the-inc/

    work page 2021

  46. [46]

    J. H. Wilkinson, The QR Algorithm for Real Symmetric Matrices with Multiple Eigenvalues, The Computer Journal 8 (1965) 85–87. URL: https://academic.oup.com/comjnl/article-lookup/doi/10.1093/comjnl/8.1.85. doi:10.1093/comjnl/8.1.85

    work page doi:10.1093/comjnl/8.1.85 1965

  47. [47]

    F. J. Corbató, On the Coding of Jacobi’s Method for Computing Eigenvalues and Eigenvectors of Real Symmetric Matrices, Journal of the ACM 10 (1963) 123–125. URL:https://dl.acm.org/doi/10.1145/321160.321161. doi:10.1145/321160.321161

    work page doi:10.1145/321160.321161 1963

  48. [48]

    L. N. Trefethen, D. Bau, Numerical Linear Algebra, 1 ed., Society for Industrial and Applied Mathematics, 1997

    work page 1997

  49. [49]

    Baiges, R

    J. Baiges, R. Codina, S. Idelsohn, Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier–Stokes equations, International Journal for Numerical Methods in Fluids 72 (2013) 1219–1243. URL:https://onlinelibrary. wiley.com/doi/10.1002/fld.3777. doi:10.1002/fld.3777

    work page doi:10.1002/fld.3777 2013

  50. [50]

    G. W. Stewart, Matrix Algorithms: V olume II: Eigensystems, Society for Industrial and Applied Mathematics, 2001. URL:http://epubs. siam.org/doi/book/10.1137/1.9780898718058. doi:10.1137/1.9780898718058

    work page doi:10.1137/1.9780898718058 2001

  51. [51]

    J. R. Magnus, H. Neudecker, Matrix differential calculus with applications in statistics and econometrics, Wiley series in probability and statistics, 3rd ed ed., Wiley, Hoboken (N.J.), 2019

    work page 2019

  52. [52]

    Unitary equivalence and normal matrices, in: C. R. Johnson, R. A. Horn (Eds.), Matrix Analysis, Cambridge Uni- versity Press, Cambridge, 1985, pp. 65–118. URL:https://www.cambridge.org/core/books/matrix-analysis/ unitary-equivalence-and-normal-matrices/01907C4F4E1B326B4E6E70CB316EFEB1. doi:10.1017/CBO9780511810817.004

    work page doi:10.1017/cbo9780511810817.004 1985

  53. [53]

    Sanderse, Energy-conserving discretization methods for the incompressible Navier-Stokes equations:application to the simulation of wind-turbine wakes, Ph.D

    B. Sanderse, Energy-conserving discretization methods for the incompressible Navier-Stokes equations:application to the simulation of wind-turbine wakes, Ph.D. thesis, Technische Universiteit Eindhoven, 2013. URL:https://research.tue.nl/en/publications/ energyconserving-discretization-methods-for-the-incompressible-navierstokes-equations--application-to-t...

    work page 2013

  54. [54]

    Sanderse, INS2D, 2018

    B. Sanderse, INS2D, 2018. URL:https://github.com/bsanderse/INS2D

    work page 2018

  55. [55]

    E. Qian, B. Kramer, B. Peherstorfer, K. Willcox, Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems, Physica D: Nonlinear Phenomena 406 (2020) 132401. URL:https://www.sciencedirect.com/science/article/pii/ S0167278919307651. doi:10.1016/j.physd.2020.132401

    work page doi:10.1016/j.physd.2020.132401 2020

  56. [56]

    Sanderse, R

    B. Sanderse, R. Verstappen, B. Koren, Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations, Journal of Computational Physics 257 (2014) 1472–1505. URL:https://linkinghub.elsevier.com/ retrieve/pii/S0021999113006670. doi:10.1016/j.jcp.2013.10.002, publisher: Elsevier BV

    work page doi:10.1016/j.jcp.2013.10.002 2014

  57. [57]

    Perron, Zur Theorie der Matrices, Mathematische Annalen 64 (1907) 248–263

    O. Perron, Zur Theorie der Matrices, Mathematische Annalen 64 (1907) 248–263. URL:http://link.springer.com/10.1007/ BF01449896. doi:10.1007/BF01449896

    work page doi:10.1007/bf01449896 1907

  58. [58]

    Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzung der physikalisch-mathematischen Classe (1912) 456–477

    G. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzung der physikalisch-mathematischen Classe (1912) 456–477

    work page 1912

  59. [59]

    F. R. Gantmakher, Applications of the theory of matrices, Dover Publications, Mineola, N.Y ., 2005. OCLC: 58720731

    work page 2005

  60. [60]

    Nikiforov, Chromatic number and spectral radius, Linear Algebra and its Applications 426 (2007) 810–814

    V . Nikiforov, Chromatic number and spectral radius, Linear Algebra and its Applications 426 (2007) 810–814. URL:https:// linkinghub.elsevier.com/retrieve/pii/S0024379507002704. doi:10.1016/j.laa.2007.06.005. 23

    work page doi:10.1016/j.laa.2007.06.005 2007