{"id":"d92c4f85-3933-4af9-9653-22cc54700f82","arxiv_id":"2411.12108","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Entropy-stable flux reconstruction is stable for under-resolved turbulent Taylor-Green and decaying isotropic turbulence simulations, matches reference spectra, and allows larger explicit time steps than over-integrated discontinuous Galerkin.","lead":"The authors test entropy-stable flux reconstruction solvers on turbulent flow benchmarks and find they are stable and accurate for implicit large eddy simulation without extra numerical dissipation. The scheme also allows larger time steps than classical discontinuous Galerkin, which could make high-order turbulence simulation cheaper.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 256^3 p7 NSFR 'DNS' reference in §3.1.2 is not grid-converged (the paper admits peak enstrophy is slightly low), yet it anchors all under-resolved ILES accuracy and SGS-model comparisons; a higher-resolution run is needed to confirm the accuracy claims.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the 256^3 p7 NSFR run is used as a DNS reference without a grid-convergence study, despite the paper's own admission that peak enstrophy is slightly lower than expected and may require higher DOF. I agree with this assessment and add a concrete downstream consequence: the under-resolved accuracy comparisons in Section 3.1.2, including the conclusion that SGS models are not beneficial because the baseline already shows a high-wavenumber TKE deficit, are all made relative to this reference. If the reference is under-resolved, those conclusions could change. This is a verification gap rather than an internal inconsistency, and it does not undermine the stability result, the CBC validation, or the measured CFL advantages. The reader's CONDITIONAL verdict is therefore appropriate, and my stress test does not move it. No fatal flaw was found; the paper's strengths include external DNS and experimental comparisons, reproducible-looking methodology details, and a clear separation of dissipation mechanisms.","tokens_in":43344,"tokens_out":11015,"duration_ms":119214,"concrete_test":"Run the same cDG NSFR.IR-GL scheme for the TGV at p=7 with 24^3 elements (192^3 numerical DOF, 168^3 unique DOF) to t*=20 and compare peak enstrophy, epsilon(t), and E(kappa,t*=9) against the 256^3 p7 result and the Dairay 512^3 reference. If peak enstrophy and the high-kappa TKE spectra change by more than about 2% between the 192^3 and 256^3 p7 runs, the 256^3 p7 'DNS' is not converged; all Section 3.1.2 under-resolved comparisons and the SGS-model conclusion should then be re-based to a converged reference.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1.1 designates the 256^3 p7 cDG NSFR.IR-GL run as the DNS reference, and Section 3.1.2 uses it, plus its p2 projection, as the baseline for every under-resolved accuracy statement in Figs. 8-20. The only convergence evidence offered is agreement with Dairay et al. 512^3 and DeBonis 512^3 in Fig. 2 and Fig. 3(a), and the text explicitly notes that 'the peak enstrophy is slightly lower than expected, although higher DOF may be required to resolve it.' A single resolution of one scheme is not a grid-convergence study. The downstream claims are sensitive to this: the statement that the 96^3 p5 run captures all length scales up to kappa_c except a slight deficit, and the recommendation that SGS models are not beneficial because they only dissipate TKE in kappa* in (25,40) while there is 'already a deficit' relative to the DNS, would both shift if the reference has more high-wavenumber TKE or a higher enstrophy peak. Thus the accuracy half of the central claim is calibrated against an unquantified reference error; stability and time-step advantages are not affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":43594,"tokens_out":11040,"duration_ms":121494,"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":[{"comment":"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.","section":"§3.1.1, §3.1.2"},{"comment":"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.","section":"Table 2, Eq. (103)"},{"comment":"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.","section":"§3.1.8, §4"}],"minor_comments":[{"comment":"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.","section":"§3.1.1, Fig. 4(b)"},{"comment":"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.'","section":"§2.9.2"},{"comment":"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.","section":"§3.1.2"},{"comment":"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.","section":"§3.1.7, Fig. 17(a)"}],"recommendation":"major_revision","confidential_remarks":"The paper is built on prior work by the same group, and the references to that work are appropriate for the method; the novelty here is the empirical assessment for turbulent flow. The main obstacle to acceptance is the lack of a grid-convergence study for the p7 DNS reference, which anchors most of the accuracy claims. The κc inconsistency in Table 2 is a concrete, fixable error. If the authors add resolution/convergence evidence and correct the cutoff issue, the paper would likely be acceptable for publication; in its current form, the accuracy half of the central claim is not yet fully supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper as the first serious test of entropy-stable flux reconstruction (NSFR) on viscous free-shear turbulence, and the news is mostly good. The scheme runs stably without over-integration or upwinding, and c+ allows about 2.6x the CFL limit of over-integrated strong DG with essentially the same results. The CBC agreement is real: their TKE error is far below the published FDS and CPR results, and they show a quantitative cost advantage for the split-form implementation.\n\nWhat's actually new: the pressure-dilatation oscillation story (collocated vs uncollocated, and the effect of Roe-type upwinding), the oversampling requirement for TKE spectra, and the CFL comparison across correction parameters. The NSFR framework itself comes from prior papers by the same group, but the LES evidence here is new.\n\nThe paper is careful. They cross-check against Dairay, DeBonis, Carton de Wiart, Van Rees, and the CBC experiment. They test many scheme variants and dissipation mechanisms. The stability claim is supported by multiple resolutions and the cost scaling is credible. No fatal flaw.\n\nThe soft spots are real but not load-bearing. The 256^3 p7 run is used as the DNS reference without a grid-convergence study; the authors themselves note the peak enstrophy is slightly low. That undercuts the quantitative accuracy claims slightly — under-resolved TKE \"deficits\" and the \"SGS models not beneficial\" recommendation are calibrated against that reference. A second resolution at 512^3 or even a coarser-finer pair would settle it. The de-aliasing comparison also confounds de-aliasing strategy with upwind dissipation: strong DG gets Roe upwinding plus over-integration, NSFR gets neither. Cleaner to compare with the same Riemann solver. The pressure-dilatation oscillations are attributed to face-term treatment but not fully explained; that's fine as a conjecture, and they do show collocation and Roe damping both help. Finally, the PHiLiP code is mentioned but no data, run scripts, or commit hash are shipped, which makes exact reproduction harder than it should be.\n\nWho is this for? CFD practitioners working on high-order methods for implicit LES, especially in the DG/FR world. It deserves a serious referee. I'd send it to review, asking for a grid-convergence study on the DNS reference and a de-aliasing comparison that holds the Riemann solver fixed. The central claims will likely survive.","headline":"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.","tokens_in":44168,"tokens_out":1890,"would_cite":true,"duration_ms":23281,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","76F65","76M10"],"pacs":["47.27.ep","47.11.Df"],"model":"deepseek-v4-flash","headline":"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.","keywords":["nonlinearly stable flux reconstruction","implicit large eddy simulation","Taylor-Green vortex","entropy-stable split form","discontinuous Galerkin","turbulent kinetic energy spectra","flux reconstruction correction parameter","Comte-Bellot-Corsin"],"falsifier":"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.","tokens_in":43089,"feed_emoji":"🌪️","tokens_out":6848,"duration_ms":64413,"temperature":0.7,"pith_summary":"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.","feed_headline":"Entropy-stable flux reconstruction halves cost of implicit LES","feed_subtitle":"The split-form scheme stays stable on coarse grids and matches turbulence spectra with larger time steps.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Identifies the accuracy and stability limitations of ESFR schemes on coarse TGV meshes, motivating the need for nonlinear stability.","marker":"[28]"},{"why":"Supplies the split-form nodal DG discretization with summation-by-parts property on which the NSFR volume term is built.","marker":"[30]"},{"why":"Establishes the skew-symmetric DG flux-differencing formulation that underlies the entropy-stable split form.","marker":"[38]"},{"why":"Shows that a kinetic-energy-preserving split-form DG scheme works for ILES of decaying turbulence, the direct precedent extended here.","marker":"[42]"},{"why":"Introduces the nonlinearly stable flux reconstruction framework in split form that this paper applies to turbulence.","marker":"[47]"},{"why":"Provides the provably stable FR discretization on curvilinear elements and the correction-operator construction used in the scheme.","marker":"[64]"},{"why":"Gives the sum-factorized evaluation of Hadamard products that makes the split-form scheme computationally competitive.","marker":"[66]"},{"why":"Supplies the Ismail-Roe entropy-conserving two-point flux used as the baseline convective flux.","marker":"[71]"},{"why":"Provides the DG-based TGV assessment and the oversampling procedure for computing turbulent kinetic energy spectra.","marker":"[24]"},{"why":"Supplies the 512^3 DNS reference used to verify the NSFR TGV results.","marker":"[82]"},{"why":"Provides the Comte-Bellot-Corsin experimental spectra used to validate the decaying homogeneous isotropic turbulence case.","marker":"[89]"}],"fun_headline_variants":["Implicit LES at half cost with entropy-stable flux reconstruction","Split-form NSFR matches turbulence spectra on coarse grids","Coarse-grid turbulence: NSFR stable and cost-effective","Oversampling velocity field clears TKE pile-up in NSFR","Face-term oscillations in NSFR tamed by Roe upwind dissipation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Implicit LES at half cost with entropy-stable flux reconstruction","Split-form NSFR matches turbulence spectra on coarse grids","Coarse-grid turbulence: NSFR stable and cost-effective","Oversampling velocity field clears TKE pile-up in NSFR","Face-term oscillations in NSFR tamed by Roe upwind dissipation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001088,"raw_usage":{"total_tokens":4589,"prompt_tokens":1028,"completion_tokens":3561,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":3476}},"tokens_in":644,"tokens_out":3561,"duration_ms":28596,"temperature":1.0,"reasoning_tokens":3476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:53:59.038925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the accuracy and stability limitations of ESFR schemes on coarse TGV meshes, motivating the need for nonlinear stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the split-form nodal DG discretization with summation-by-parts property on which the NSFR volume term is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the skew-symmetric DG flux-differencing formulation that underlies the entropy-stable split form."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that a kinetic-energy-preserving split-form DG scheme works for ILES of decaying turbulence, the direct precedent extended here."},{"cited_title":"Cicchino, S","cited_arxiv_id":null,"evidence_quote":"Introduces the nonlinearly stable flux reconstruction framework in split form that this paper applies to turbulence."},{"cited_title":"Cicchino, D","cited_arxiv_id":null,"evidence_quote":"Provides the provably stable FR discretization on curvilinear elements and the correction-operator construction used in the scheme."},{"cited_title":"Cicchino, S","cited_arxiv_id":null,"evidence_quote":"Gives the sum-factorized evaluation of Hadamard products that makes the split-form scheme computationally competitive."},{"cited_title":"Ismail, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Ismail-Roe entropy-conserving two-point flux used as the baseline convective flux."},{"cited_title":"Carton de Wiart, K","cited_arxiv_id":null,"evidence_quote":"Provides the DG-based TGV assessment and the oversampling procedure for computing turbulent kinetic energy spectra."},{"cited_title":"Dairay, E","cited_arxiv_id":null,"evidence_quote":"Supplies the 512^3 DNS reference used to verify the NSFR TGV results."},{"cited_title":"Comte-Bellot, S","cited_arxiv_id":null,"evidence_quote":"Provides the Comte-Bellot-Corsin experimental spectra used to validate the decaying homogeneous isotropic turbulence case."}],"review_version":1}