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REVIEW 3 major objections 4 minor 92 references

Large Eddy Simulation using Nonlinearly Stable Flux Reconstruction

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper establishes that entropy-stable flux reconstruction performs implicit large-eddy simulation of subsonic free-shear turbulence stably and accurately, at lower cost than over-integrated discontinuous Galerkin.

desk verdict Solid, careful LES study of entropy-stable flux reconstruction; the main stability and time-step claims hold, but the DNS reference needs a grid-convergence check before the accuracy comparisons are quoted as quantitative. read the letter →

arxiv 2411.12108 v1 pith:PUKJTCWQ submitted 2024-11-18 physics.flu-dyn cs.NAmath.NAphysics.comp-ph

Julien Brillon , Siva Nadarajah This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NAphysics.comp-ph MSC 65M6076F6576M10 PACS 47.27.ep47.11.Df
keywords nonlinearlystablefluxreconstructionimplicitlargeeddysimulationTaylor-Greenvortexentropy-stablesplitformdiscontinuousGalerkinturbulentkineticenergyspectracorrectionparameterComte-Bellot-Corsin
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 tries to establish that nonlinearly stable flux reconstruction (NSFR) in split form is a practical implicit large-eddy-simulation (ILES) method for subsonic free-shear turbulence. It argues that, unlike classical discontinuous Galerkin, NSFR is stable on arbitrarily coarse grids without upwind dissipation, over-integration, or a sub-grid-scale model, and that its entropy-conserving formulation still reproduces DNS and experimental turbulence spectra. It further claims that increasing the flux-reconstruction correction parameter to its largest stable value permits explicit time steps about 2.6 times larger than over-integrated strong DG, making the scheme about twice as cheap for similar accuracy. The study also reports that the two-point numerical flux choice does not affect the solution, that standard eddy-viscosity SGS models do not help, and that oversampling the velocity field is necessary to avoid an artificial pile-up of turbulent kinetic energy near the spectral cut-off. A sympathetic reader would care because the results point to a route toward cheap, robust high-order LES that does not depend on ad hoc stabilization.

What carries the argument

The load-bearing object is the NSFR split-form discretization: the convective term uses a two-point entropy-conserving flux evaluated at pairs of nodes and combined through a skew-symmetric stiffness operator, which yields discrete entropy stability without over-integration. The correction parameter c in the ESFR correction operator enters through a modified mass matrix that acts as a linear filter on the DG residual; c = cDG recovers DG, while c = c+ (the largest stable value) damps the highest mode and raises the explicit CFL limit. Sum-factorized tensor and Hadamard products make the two-point flux evaluation cheap enough that the whole scheme runs at about half the cost of over-integrated DG.

What would settle it

Run a grid-convergence study for the TGV case with the p7 NSFR scheme at $128^{3}$, $256^{3}$, and $512^{3}$ equivalent degrees of freedom; if the peak enstrophy and dissipation curves still shift between the two finest resolutions, the reference DNS is not converged and the under-resolved comparisons inherit its error.

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Extended reading notes

Core claim

The paper's central claim is that NSFR in split form is a viable ILES scheme for subsonic free-shear turbulence: it is entropy-stable on every under-resolved grid tested with no added dissipation, it matches DNS and experimental turbulent kinetic energy spectra when the velocity field is oversampled, and choosing the correction parameter c+ yields solutions close to those of cDG and strong DG at about half the computational cost. The authors also claim that the spurious pressure-dilatation oscillations seen in the raw diagnostic are a face-term artifact, significantly weaker for a collocated scheme and effectively eliminated by Roe-type upwind dissipation, rather than a defect in the physical solution. The choice of two-point numerical flux does not affect the result, and standard eddy-viscosity-based SGS models do not improve the under-resolved TGV results.

Load-bearing premise

The $256^{3}$ p7 NSFR run is treated as a converged DNS reference even though no grid-convergence study is presented, and the paper acknowledges the peak enstrophy is slightly lower than expected.

Editorial extensions

If this is right

  • For Taylor-Green-type subsonic free-shear turbulence, NSFR in split form is stable at every under-resolved resolution tested with no added dissipation, so implicit LES does not need over-integration or SGS models for stability.
  • Setting the correction parameter to c+ instead of cDG raises the physically consistent CFL limit to 0.36, about 2.6 times the 0.14 allowed by strong DG with Roe dissipation and over-integration.
  • At polynomial degrees p = 5 to p = 12, the split-form NSFR costs about half the CPU time per time step of the over-integrated strong DG scheme on the same TGV setup.
  • Turbulent kinetic energy spectra must be computed from a velocity field oversampled to 2(p+1) nodes per direction; without oversampling an apparent TKE pile-up appears near the cut-off and the integrated kinetic-energy error grows.
  • A collocated NSFR scheme with Roe-type upwind dissipation removes the spurious pressure-dilatation oscillations, indicating that the face-term treatment, not the volume scheme, is the source of that artifact.

Reading between the lines

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

  • The stability-on-any-grid result suggests a testable extension: run NSFR on even coarser TGV grids or on a wall-bounded transitional flow and see whether split-form entropy stability alone keeps the solution bounded; if it does, over-integration could be dropped from production ILES codes.
  • The paper's face-term explanation for pressure-dilatation oscillations implies that other entropy-stable DG and FR codes should show the same pattern, so comparing face-flux interpolation strategies across codes would isolate the artifact's true source.
  • The oversampling requirement implies that spectral cut-off comparisons between high-order schemes are only meaningful if all schemes are sampled above their element polynomial content; otherwise reported pile-up may be a post-processing artifact rather than a scheme property.
  • Because c+ already allows a 2.6 times larger explicit time step than DG, NSFR may be especially attractive for under-resolved simulations on many-core machines where explicit time integration dominates, since the reduced step count can outweigh the split form's extra per-step work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents an extensive assessment of nonlinearly stable flux reconstruction (NSFR) schemes for direct numerical simulation and implicit large-eddy simulation of subsonic, shock-free turbulent flows. The main test cases are the viscous Taylor-Green vortex at Re=1600 and the Comte-Bellot-Corsin decaying homogeneous isotropic turbulence experiment. The authors verify a p7 cDG NSFR scheme as a DNS reference against independent data from Dairay, DeBonis, and Carton de Wiart, then use that reference to judge under-resolved ILES results at lower resolutions. They compare split-form NSFR with classical over-integrated DG, investigate the FR correction parameter c, collocated versus uncollocated flux nodes, two-point flux choices, added Riemann dissipation, and sub-grid-scale models, and they demonstrate the necessity of oversampling for TKE spectra. The central claims are that entropy-stable split-form NSFR is stable without over-integration, permits larger explicit time steps than over-integrated DG, is more cost-effective when implemented with sum factorization, and provides accurate implicit LES for this class of flows, while classical eddy-viscosity SGS models do not improve the results.

Significance. If the claims hold, the paper makes a useful contribution to high-order CFD for turbulent flows: it provides evidence that a provably nonlinear stable FR scheme can serve as an ILES method without extra de-aliasing or upwind dissipation, and it quantifies a time-step advantage over classical DG. The work is strengthened by comparisons to multiple independent references (Dairay, DeBonis, Carton de Wiart, CBC experiment), by the availability of the open-source PHiLiP implementation, and by the explicit analysis of post-processing issues such as oversampling and the pressure-dilatation diagnostic. The main significance is therefore an application-oriented validation of NSFR for free-shear turbulence, rather than a new theoretical result. The assessment is tempered by the fact that the DNS reference used for all under-resolved accuracy comparisons is not shown to be grid-converged, and one reported cut-off wavenumber is inconsistent with the formula in the paper.

major comments (3)
  1. [§3.1.1, §3.1.2] The 256^3 p7 cDG NSFR.IR-GL result is designated as the DNS reference and is used from §3.1.2 onward, but no grid-convergence study supports this designation. The manuscript itself states in §3.1.1 that 'the peak enstrophy is slightly lower than expected, although higher DOF may be required to resolve it.' Every under-resolved accuracy statement in Figs. 8–20, including the statement that the 96^3 p5 run captures all length scales up to κc with a slight deficit, and the recommendation in §3.1.8 that SGS models are not beneficial because there is 'already a deficit' relative to the DNS, depends on the high-wavenumber content and enstrophy of this reference. If the reference is under-resolved, the apparent spectral deficits could be underestimated and the SGS conclusions could change. Please add a grid-convergence study (e.g., a second resolved resolution or a systematic comparison against the 512^3 pseudo-spectral data for all quantities of interest) and quantify the uncertainty of the reference before using it as the baseline for ILES accuracy judgments.
  2. [Table 2, Eq. (103)] The p7 row of Table 2 reports κc = 122, but Eq. (103) with Nel = 32 and p = 7 gives Nel p / 2 = 32 × 7 / 2 = 112. This is not a bookkeeping detail: the DNS spectra in Figs. 5 and 6 and the oversampling errors in Table 4 are plotted or integrated up to this cut-off. If the intended cutoff is 112, then the statement that the p7 DNS 'captures all length scales in the flow up to κc' and the integrated-energy verification need to be recomputed. Please correct the table, the figures, and any derived errors, or explain explicitly why a different cutoff definition is being used for the p7 case.
  3. [§3.1.8, §4] The conclusion that standard eddy-viscosity SGS models do not improve NSFR for transitional free-shear flows is drawn from the TGV case at 96^3 DOF and is explicitly based on the observation that the baseline already has a TKE deficit relative to the DNS at κ* ∈ (25,40). Because that DNS reference is not demonstrated to be converged, the conclusion is not robust. If the reference DNS has additional high-wavenumber energy, the baseline deficit may be smaller than reported and the SGS models may have a positive effect. Please either soften this conclusion to the resolution and reference used, or test it against a converged reference at a higher resolution before making a general recommendation against SGS models for NSFR.
minor comments (4)
  1. [§3.1.1, Fig. 4(b)] The discussion of effective viscosity contains a likely wording error: the text says that for t* > 5 the effective viscosity is greater than 1 for all schemes except the p3 uncollocated dissipation-free scheme, and then says that 'only this scheme was ever more dissipative than it should be.' If ν_effective > 1 means more dissipative, then the exception should be the scheme that is not more dissipative; please clarify whether 'more' should read 'less' or revise the sentence.
  2. [§2.9.2] There is a typographical error in the sentence introducing the Roe flux: 'The the eigenvalues or wave speeds' should read 'The eigenvalues or wave speeds.'
  3. [§3.1.2] The phrase 'the entropy conserving scheme is stable for any grid without any added dissipation' is stronger than what the numerical experiments alone can show, since only a small set of Cartesian grids was tested. If the statement is intended as a consequence of the nonlinear stability proofs from the cited prior work, that logical connection should be made explicit in this section.
  4. [§3.1.7, Fig. 17(a)] The Lax-Friedrichs upwind scheme is described as exhibiting 'anti-dissipative behaviour' because it produces a higher peak enstrophy than the projected DNS; given that LxF is usually dissipative, this is surprising and should be explained in the text, for example by pointing to the sign of the upwind term noted in §2.9.1.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: NSFR stability and cost claims are benchmarked against independent references; same-group citations provide background theory, not fitted predictions.

full rationale

No circular step can be exhibited by quotation and reduction. The NSFR discretization is imported from prior same-group papers ([47,64,65]), and the paper's stability statements are provable properties of that framework with stated assumptions, not parameters fit to the present data; under the review rules this counts as independent support. The empirical validation anchors are external: Dairay et al. 512^3, DeBonis 512^3, Carton de Wiart 512^3, Van Rees pseudo-spectral data, and the CBC experiment. The 'DNS' reference for the under-resolved section is the paper's own 256^3 p7 cDG NSFR.IR-GL run, and the paper explicitly concedes in Section 3.1.1 that 'the peak enstrophy is slightly lower than expected, although higher DOF may be required to resolve it for this given scheme.' Using that single-resolution run as the baseline for all under-resolved comparisons is a grid-convergence and validation caveat, not a circularity, because that run was itself cross-checked against independent DNS references. Similarly, the CFL-limit and CPU-time comparisons in Section 3.1.5 are measured quantities, and the statement that 'by using c+ we can obtain a solution similar to that of cDG and strong DG at about half the computational cost' is a cost/accuracy comparison, not a fitted prediction. Minor self-citations to the same group's NSFR framework are present, which is why the score sits at 2 rather than 0, but they are not load-bearing in the derivation of the paper's central ILES conclusions.

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

The central ILES findings are empirical and do not require a fitted free parameter. The listed constants are standard model and algorithm choices from prior literature. The main unproven background is the entropy-stability proof of NSFR and the assumed convergence of the p7 DNS reference.

free parameters (5)
  • Smagorinsky coefficient C_S = 0.10
    Used in all Smagorinsky-type SGS models in Section 2.6.1; chosen from Deardorff literature, not calibrated to this flow, but the SGS conclusions depend on it.
  • High-pass filter polynomial degree pL = 3
    Used for HPF SGS models and the dynamic model test filter in Sections 2.6.3 and 2.6.4; chosen by hand.
  • Dynamic Smagorinsky test-filter degree pTF = 3
    Set in Section 2.6.4 for the dynamic model; not calibrated within this paper.
  • Turbulent Prandtl number Pr_t = 0.6
    Set in Section 2.4 following reference [54]; affects SGS heat flux, not the central solution accuracy.
  • FR correction parameter c = cDG, c+, cSD, cHU
    Varied in Section 3.1.5; the c+ case gives the reported CFL advantage. Values come from prior stability analyses, not fitted to TGV data.
assumptions (5)
  • domain assumption The compressible Navier-Stokes equations with ideal gas, Sutherland viscosity, Stokes hypothesis, and standard Prandtl number model the flows studied.
    Used as the governing model in Sections 2.1 and 2.2; the paper does not validate this modeling choice.
  • standard math The NSFR discretization in Eqs. (70) and (71) is provably entropy stable as established in cited prior work.
    The paper relies on the entropy-stability proofs of Cicchino et al. and Chan without reproving them.
  • domain assumption The 256^3 p7 NSFR result is treated as a converged DNS reference.
    Section 3.1.1 uses this run as the DNS reference; no grid-convergence study is shown, and the paper acknowledges the peak enstrophy may need higher DOF.
  • domain assumption The CBC experimental spectrum at tV/M=42 can be used as a numerical initial condition through synthetic turbulence generation.
    Section 3.2.1 initializes the DHIT case using TurboGenPY; synthetic generation may introduce modes not present in the experiment.
  • domain assumption The projected DNS from the 256^3 p7 result onto 96^3 p2 is a valid comparison baseline for under-resolved runs.
    This projection is used throughout Sections 3.1.2 through 3.1.8 as the reference for coarse-grid accuracy.

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Pith. "Pith review of Large Eddy Simulation using Nonlinearly Stable Flux Reconstruction." pith.science (2026). https://pith.science/paper/PUKJTCWQ

@misc{pith2026241112108,
  author       = {Pith},
  title        = {Pith review of: Large Eddy Simulation using Nonlinearly Stable Flux Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUKJTCWQ}},
  note         = {Machine review of arXiv:2411.12108}
}
read the original abstract

The performance of the nonlinearly stable flux reconstruction (NSFR) schemes for resolving subsonic viscous turbulent free-shear flows is investigated. The schemes are extensively verified for the direct numerical simulation (DNS) of the Taylor-Green Vortex (TGV) problem. Several under-resolved simulations of the TGV problem are conducted to assess the performance of NSFR for large eddy simulation that is implicitly filtered and fully implicit (ILES). Increasing the flux reconstruction correction parameter ensures that NSFR is stable and accurate for ILES while allowing for larger explicit time-steps. The entropy-stable schemes implemented with sum-factorization for tensor and Hadamard products are shown to be more cost-effective than classical DG with over-integration. The choice of the two-point (TP) numerical flux does not impact the solution and the use of standard eddy-viscosity-based sub-grid scale models does not yield improvements for the problem considered. From the DNS results, the pressure dilatation-based dissipation rate for the nonlinearly stable schemes is consistent with literature when computed from the kinetic energy (KE) budget terms, while spurious oscillations are seen when the term is directly computed. The magnitude of these oscillations is significantly lower for a collocated scheme and are effectively eliminated with the addition of Roe upwind dissipation to the TP numerical flux. Therefore, these oscillations are believed to be associated with the treatment of the face terms in nonlinearly stable schemes. It is shown that oversampling the velocity field is necessary for obtaining accurate turbulent KE (TKE) spectra and eliminates an apparent pile-up of TKE at the smallest resolved scales. Lastly, the TKE spectra for a decaying homogeneous isotropic turbulence case are in good agreement with experiment measurements and computational results in the literature.

Figures

Figures reproduced from arXiv: 2411.12108 by the authors.

Figure 1
Figure 1. Isocontours of pressure coloured by Mach number for the viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Temporal evolution of kinetic energy, enstrophy, dissipation rate, and dissipation components for the viscous TGV [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Temporal evolution of observed and computed pressure dilation for the viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Temporal evolution of pressure dilatation and effective viscosity for the viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Nondimensional TKE spectra of viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: Nondimensional TKE spectra of viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: Contours of nondimensional vorticity magnitude [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: Temporal evolution of dissipation rate and enstrophy for the viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: Nondimensional turbulent kinetic energy spectra at [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: Filled contours of vorticity magnitude |ω| at time t ∗ = 9 in the plane x ∗ = 0 (one quadrant shown due to symmetry) for the viscous TGV at Re∞ = 1600 using p5 cDG NSFR.IR-GL with 963 DOF 3.1.3. De-aliasing strategies High-order methods for ILES are prone to aliasing …
Figure 11
Figure 11. Figure 11: Temporal evolution of enstrophy, dissipation rate, and pressure dilatation along with the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 12
Figure 12. Figure 12: Scaling of CPU time for one time step of viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]
Figure 13
Figure 13. Figure 13: Temporal evolution of dissipation rate, enstrophy, and pressure dilatation along with the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p037_13.png]
Figure 14
Figure 14. Figure 14: Temporal evolution of dissipation components and pressure dilatation, along with the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p039_14.png]
Figure 15
Figure 15. Figure 15: Temporal evolution of enstrophy and the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p041_15.png]
Figure 16
Figure 16. Figure 16: Viscous TGV at Re∞ = 1600 with 963 DOF at p5 for uncollocated cDG NSFR with different two-point numerical fluxes 3.1.7. Added Riemann Solver Dissipation To investigate the effects of adding different Riemann solver dissipation, i.e. the upwinding term, to the two-poin…
Figure 17
Figure 17. Figure 17: Temporal evolution of enstrophy and pressure dilatation, along with the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p043_17.png]
Figure 18
Figure 18. Figure 18: Temporal evolution of enstrophy and pressure dilatation for the viscous TGV at Re [PITH_FULL_IMAGE:figures/full_fig_p044_18.png]
Figure 19
Figure 19. Figure 19: Temporal evolution of dissipation rate, enstrophy and pressure dilatation, along with the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p046_19.png]
Figure 20
Figure 20. Figure 20: Temporal evolution of dissipation components and the TKE spectra at [PITH_FULL_IMAGE:figures/full_fig_p047_20.png]
Figure 21
Figure 21. Figure 21: Flow field initialization for the initial Comte-Bellot and Corsin (CBC) experiment using NSFR with 128 [PITH_FULL_IMAGE:figures/full_fig_p049_21.png]
Figure 22
Figure 22. Figure 22: Nondimensional TKE spectra of DHIT using p3 [PITH_FULL_IMAGE:figures/full_fig_p051_22.png]

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