Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Can Explicit Subgrid Models Enhance Implicit LES Simulations? A Very High-Order Solver Perspective

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Explicit Vreman subgrid-scale modeling helps very-high-order implicit LES only when the flow is under-resolved; in well-resolved simulations the scheme's own split-form and Riemann-solver dissipation already suffices, and adding the model c

desk verdict Careful P=7 TGV parameter study with a useful regime map; abstract overclaims a lower-order comparison that the body never makes. read the letter →

arxiv 2512.04574 v2 pith:I4LHGQSR submitted 2025-12-04 physics.flu-dyn cs.NAmath.NA

Gonzalo Rubio , Gerasimos Ntoukas , Miguel Chávez-Módena , Oscar Mariño , Bernat Font , Oriol Lehmkuhl , Eusebio Valero , Esteban Ferrer This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA PACS 47.11.Fg47.27.Ep
keywords discontinuousGalerkinspectralelementmethodimplicitLESVremansubgrid-scalemodelTaylor-GreenvortexsplitformRiemannsolverfidelityGPUhigh-ordercomputing
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

This paper asks whether adding an explicit Vreman subgrid-scale model to a very high-order discontinuous Galerkin solver improves implicit large-eddy simulations. Using the Taylor-Green vortex at Reynolds 1600 and in the inviscid limit, it finds the answer is regime-dependent: when the mesh resolves the flow, split-form stabilization plus Riemann-solver dissipation is already enough, and the Vreman model is neutral or harmful; when resolution is too coarse, a weak Vreman term removes excess high-wavenumber energy and improves spectral accuracy. The paper identifies the configuration that performed best in each regime—Chandrasekhar split form with Roe flux and no SGS for transition, LD-Roe with Cv=0.01 for well-resolved turbulence, Roe with Cv=0.07 for under-resolved turbulence—and concludes that no static setting spans all regimes. The work matters because GPU-oriented computing favors very high polynomial orders, where tuning the balance between numerical and modeled dissipation becomes the main accuracy lever.

What carries the argument

The carrying mechanism is the Vreman eddy-viscosity SGS model with the element-based filter length Δ = V^(1/3)/(P+1), whose single constant Cv controls both the amount of added dissipation and the wavenumber at which that dissipation activates. It is placed inside a DGSEM discretization that already has two intrinsic dissipation sources: split-form stabilization (using the Chandrasekhar form for robustness) and Riemann solvers (Roe and its low-dissipation variant LD-Roe). The diagnostics—kinetic-energy dissipation rate over time and the energy spectrum at t/tc=9—expose which of the three dissipation sources dominates in each regime, and the spectral difference E_base(k) - E_SGS(k) locates th

What would settle it

Run the same Re=1600 Taylor-Green vortex at a lower polynomial order (say P=3 or P=4) with the same total degrees of freedom and compare the spectral accuracy of iLES versus iLES+Vreman; the paper predicts SGS is neutral or harmful there, so a clear improvement would falsify the regime map. A second check is to scan Cv finely in 0.01–0.07 in the inviscid case: the paper's trade-off implies a non-monotone accuracy curve, so monotone improvement with Cv would contradict it.

Watch

Extended reading notes

Core claim

The central discovery is a regime map for dissipation in very high-order DG. In the well-resolved TGV at Re=1600 with P=7, the inherent dissipation from split forms and Riemann solvers matches the reference transitional dynamics, and the Vreman model's added viscosity acts in a wavenumber range that overlaps the scheme's own dissipation, so it does not improve accuracy and the larger constant (Cv=0.07) visibly over-damps intermediate scales. In the turbulent phase of that case, a low-dissipation Riemann flux (LD-Roe) combined with a weak Vreman constant (Cv=0.01) gives the best high-wavenumber spectrum. In the inviscid, strongly under-resolved TGV, the same weak model is insufficient: Roe wi

Load-bearing premise

The regime map is built solely on the P=7 Taylor-Green vortex on a 16^3 mesh with two Vreman constants; the abstract announces lower-order comparisons at equal degrees of freedom, but the only reported lower-order data is a GPU-efficiency benchmark, not a flow-accuracy comparison. If P=7 TGV does not represent other very-high-order DG set-ups, the practical guidance does not transfer.

Editorial extensions

If this is right

  • In well-resolved very-high-order LES, explicit SGS modeling can be omitted: the split-form plus Roe configuration matches the transitional reference, and adding Vreman only shifts dissipation into scales the scheme already handles.
  • For the turbulent phase of a well-resolved simulation, the most accurate tested setup combines a low-dissipation Riemann solver with a weak Vreman constant (Cv=0.01), which resolves the energy pile-up without over-damping intermediate wavenumbers.
  • For strongly under-resolved flows, a larger Vreman constant (Cv=0.07) is needed to remove high-wavenumber energy, but it still over-damps the scales just below the cutoff, so the correct constant depends on how under-resolved the simulation is.
  • The same split form with the same SGS model can be the best or the worst choice depending on the flow regime, so a static numerical configuration cannot be optimal across a simulation that passes through laminar, transitional, and turbulent phases.
  • Scale-aware or adaptive dissipation—the paper points to spectral vanishing viscosity and data-driven tuning—becomes a necessary next step rather than an optional refinement.

Reading between the lines

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

  • The paper does not test lower-order TGV flows at equal degrees of freedom despite announcing them; if a lower-order run showed Vreman improving a well-resolved LES, the regime map would not generalize. That comparison is the most direct untested extension.
  • If the constant-to-wavenumber relation holds generally, it yields a practical calibration rule: run the iLES baseline briefly, find the wavenumber where energy piles up, and choose Cv so the model's activation wavenumber sits just below it. The paper demonstrates the relation but does not codify the rule.
  • A natural way to get scale-selective dissipation without a global constant is p-adaptivity: locally lowering the polynomial order damps near-cutoff scales more strongly, which mimics the weak-SGS effect the paper found beneficial in under-resolved regions.
  • Because only Cv=0.01 and Cv=0.07 were scanned, the spectral evidence suggests an intermediate constant—or a wavenumber-dependent variant—might hit the sweet spot of removing the pile-up without flattening intermediate scales; the paper notes the optimum may lie between the two values but does not scan it.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies how an explicit Vreman subgrid-scale model interacts with the inherent numerical dissipation of a high-order DGSEM solver for the Taylor–Green vortex. Using a fixed 16^3 mesh and polynomial order P=7, the authors compare split-form stabilization, Riemann solvers (central, Roe, LD-Roe, matrix dissipation, Lax–Friedrichs), and Vreman constants C_v=0.01 and 0.07 in both a viscous (Re=1600) and an inviscid (Re=∞) configuration. On the basis of kinetic-energy dissipation-rate histories and energy spectra at t/t_c=9, they conclude that the usefulness of explicit SGS modeling is regime-dependent: in well-resolved LES the implicit dissipation of split forms and Riemann solvers is sufficient, while under strongly under-resolved conditions a weak SGS contribution can remove excess high-wavenumber energy, but a large constant over-damps intermediate scales. The paper also includes a GPU-efficiency benchmark showing that P=7 approximately doubles throughput per DOF relative to P=3.

Significance. If the regime map is accepted, the paper provides useful practical guidance for choosing dissipation mechanisms in very-high-order DG turbulence simulations on GPU architectures. The systematic comparison of many split forms, Riemann fluxes, and two Vreman constants against high-resolution reference data is a strength, as is the use of the publicly available HORSES3D solver, which supports reproducibility. However, the central generalization to lower polynomial orders at matched degrees of freedom is not supported by any lower-order TGV simulation, and the abstract's blanket statement that Vreman modeling does not improve accuracy in well-resolved cases is contradicted by the paper's own conclusions. The relevance of the findings to the broader practical guidance therefore rests on a single flow, a single mesh, a single polynomial order, and qualitative visual comparisons.

major comments (3)
  1. [Abstract; §3.1; §5] The abstract claims the study is 'comparing lower- and very high-order configurations at similar degrees of freedom,' and §5 states that 'this issue is less pronounced at lower orders, where the inherent numerical damping partially compensates for the deficiencies of the model.' These cross-order claims are not supported by any experiment in the paper: all TGV simulations use a single 16^3 mesh and P=7 (§3.1), and the only P=3–7 data are the GPU throughput measurements in Fig. 1, which concern computational efficiency, not spectral fidelity or dissipation. The load-bearing condition for the abstract's practical guidance is that P=7 behavior is representative of very-high-order DG and that lower-order behavior at matched DOF follows the stated ordering. Since that condition is never tested, the generalization is an assertion. Either lower-order TGV runs at matched DOF must be added, or th
  2. [Abstract vs §6; §4.1.5] The abstract states, 'In the well-resolved cases considered, Vreman modeling does not improve accuracy because its active wavenumber range overlaps with the scheme's inherent dissipation.' This is internally inconsistent with the conclusions in §6, which recommend, for well-resolved LES in the turbulent regime, 'superior spectral fidelity is obtained using the split form and an LD-Roe flux, supplemented by an SGS model with a low constant (Cv = 0.01).' The same recommendation is supported by §4.1.5, where 'LD-Roe coupled with the Vreman model provides the best spectral fidelity.' These are not merely wording differences: one central message says Vreman is not helpful in well-resolved cases, while the other says the best well-resolved turbulent configuration uses Vreman. The abstract and conclusions need to be reconciled.
  3. [§4, Figs. 3–10] The accuracy assessments are made exclusively by visual inspection of dissipation-rate curves and energy spectra, without any quantitative error measure. This matters because the conclusions are finely graded: e.g., §4.1.3 claims that C_v=0.07 'over-dissipates energy at intermediate scales' while C_v=0.01 leaves the highest wavenumbers 'slightly under-dissipated'; §4.2.2 claims that C_v=0.01 'overestimates energy at intermediate wavenumbers.' Such claims are load-bearing for the proposed regime map and for the guidance in §6, but they rest on subjective comparison to reference spectra. The authors should provide quantitative metrics, for example relative L1/L2 errors of E(k) over defined wavenumber ranges or time-integrated dissipation-rate errors, so that 'best spectral fidelity' and 'over-dissipation' are defined operationally.
minor comments (5)
  1. [§4.1.2] The first sentence of §4.1.2 contains a typo: 'As in the reminder of the paper' should be 'As in the remainder of the paper.'
  2. [Fig. 3 caption] The caption says 'The standard and Morinishi schemes are shown only in Fig. 3a, as they became unstable before t/t_c = 9. Results for all schemes are shown in both figures' — this is self-contradictory. If standard and Morinishi are omitted from Fig. 3b, the sentence should say 'Results for all stable schemes are shown in both figures.'
  3. [§2.6; Fig. 1] The GPU efficiency metric Time/(DOF×RHS) is introduced without a definition of the RHS evaluation context (e.g., which flux/split form). Since the comparison spans P=3–7, a one-sentence statement of the test problem used for the benchmark would improve clarity.
  4. [§4.1.1] The standard versus split-form comparison uses Gauss nodes for the standard discretization and Gauss–Lobatto nodes for the split forms, so the effect of split-form stabilization is not isolated from the effect of nodal distribution. The text notes this, but a brief interpretive caution would be helpful for readers.
  5. [§6] The sentence 'Note that as only C_v = 0.01 and C_v = 0.07 were tested, the global optimum for this specific problem may lie within this range' is appropriate, but the bullet list should make explicit that the 'optimal' labels follow from only two constants, not from a true optimization over C_v.

Circularity Check

1 steps flagged · score 1.0 of 10

Numerical experiment with external references; not circular, though the broad regime claims are only weakly supported by a single P=7 case

  1. renaming known result [Sections 4.1.3 and 4.2.2 (Figures 9–10); conclusion in Section 6]
    "increasing the constant to the typical finite-volume value of Cv = 0.07 aligns the spectrum around k = 20, but results in over-dissipation for k > 20."

    The two-point sweep Cv ∈ {0.01, 0.07} is reframed as a finding about the 'optimal' constant lying in that range. This is a known-knob-tuning observation, not a derivation; it does not make the work circular because it is a posteriori description of an experiment, not a fitted input renamed as a prediction.

full rationale

The paper is a numerical experiment, not a derivation. Its target claims — that Vreman SGS is neutral in well-resolved high-order DG and helpful in under-resolved settings — are evaluated against external datasets (Bull & Jameson, Fehn et al., Carton de Wiart) rather than fitted from them. The Vreman constants are taken from the literature (Cv = 0.07 from finite-volume practice, Cv = 0.01 from high-order WRLES [31]) and only two values are swept; the paper explicitly states 'the global optimum for this specific problem may lie within this range,' so no fitted parameter is relabeled as a prediction. The abstract promises 'comparing lower- and very high-order configurations at similar degrees of freedom,' but the body only reports TGV simulations at P=7 on a fixed 16^3 mesh; the P=3–7 comparison exists only as a GPU-efficiency benchmark (Fig. 1), and the statement in Section 5 that 'this issue is less pronounced at lower orders' is asserted without supporting simulations. That is a representativeness/evidence gap, not a circularity: the conclusion is not entailed by its inputs by construction. Self-citations are present (e.g., HORSES3D [40, 38], Duan & Wang is external) but the central claim does not reduce to them. Score 1 reflects the minor, non-load-bearing gap between the abstract's lower-order framing and the actual P=7-only evidence, not a self-consistent derivation loop.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The quantitative conclusions rest on two tested Vreman constants, the SBP-SAT split-form framework, the chosen filter length, and the fidelity of external reference spectra; the paper introduces no new entities.

free parameters (1)
  • Vreman model constant C_v = 0.01 and 0.07 (tested, not fitted)
    The eddy-viscosity constant is an input taken from the literature; the central guidance on optimal SGS strength is conditional on the two tested values, and the paper acknowledges the optimum may lie in between.
assumptions (5)
  • standard math SBP-SAT split forms on Gauss-Lobatto nodes provide discrete energy/entropy stability and control aliasing.
    Invoked throughout Sec. 2.2 to justify stability of under-resolved simulations without explicit SGS.
  • domain assumption The Vreman model with filter length Delta = V^(1/3)/(P+1) is an adequate representation of unresolved-scale dissipation in this DG formulation.
    Eq. (1) and Appendix A couple mu_t to resolved velocity gradients; no a priori test of model adequacy is provided.
  • domain assumption The external reference solutions (Bull & Jameson 512^3 DRP, Fehn et al. 8192^3, Carton de Wiart 512^3 pseudospectral) are accurate enough to judge spectral fidelity.
    Used as benchmarks in Sec. 3/4 for dissipation rate and spectra; any error in these references transfers to the conclusions.
  • domain assumption The Taylor-Green vortex at Re=1600 and in the inviscid limit is representative of well-resolved and under-resolved LES regimes relevant to GPU-oriented high-order DG.
    The paper generalizes from this single test case; authors acknowledge wall-bounded flows are future work (Sec. 6).
  • standard math The BR1 viscous discretization is neutrally stable and introduces minimal dissipation.
    Stated in Sec. 2.1 and supported by cited reference [43].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Can Explicit Subgrid Models Enhance Implicit LES Simulations? A Very High-Order Solver Perspective." pith.science (2026). https://pith.science/paper/I4LHGQSR

@misc{pith2026251204574,
  author       = {Pith},
  title        = {Pith review of: Can Explicit Subgrid Models Enhance Implicit LES Simulations? A Very High-Order Solver Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I4LHGQSR}},
  note         = {Machine review of arXiv:2512.04574}
}
read the original abstract

High-order discontinuous Galerkin (DG) methods offer excellent accuracy for turbulent-flow simulations and are increasingly attractive on GPU-oriented architectures, where high polynomial orders can improve arithmetic intensity. However, very high-order under-resolved simulations remain sensitive to the balance between numerical and modeled dissipation. We investigate how explicit Vreman subgrid-scale (SGS) modeling interacts with dissipation from split-form stabilization and Riemann solvers in a DGSEM framework. Using the three-dimensional Taylor-Green vortex at Re=1600 and in the inviscid limit, we assess kinetic-energy dissipation, spectral accuracy, and stability across well-resolved, under-resolved viscous, and strongly under-resolved regimes, comparing lower- and very high-order configurations at similar degrees of freedom. The usefulness of explicit SGS modeling depends strongly on resolution, polynomial order, and the numerical dissipation already present. In the well-resolved cases considered, Vreman modeling does not improve accuracy because its active wavenumber range overlaps with the scheme's inherent dissipation. At similar degrees of freedom, lower-order simulations introduce stronger damping near the smallest resolved scales, whereas very high-order simulations preserve more spectral content but are more susceptible to high-wavenumber energy accumulation when dissipation is insufficient. Under stronger under-resolution, a weak SGS contribution can control this accumulation, while excessive SGS dissipation degrades intermediate scales. These results identify regimes in which explicit SGS modeling is beneficial, neutral, or detrimental, and provide practical guidance for selecting dissipation mechanisms in very high-order DG turbulence simulations suited to modern GPU architectures.

Figures

Figures reproduced from arXiv: 2512.04574 by the authors.

Figure 1
Figure 1. GPU efficiency of HORSES3D on a single GPU. Efficiency increases with P, roughly doubling from P = 3 to P = 7, due to higher arithmetic intensity and reduced memory traffic. Although the present work is not focused on GPU performance, we include a brief assessment for completeness. The GPU performance of HORSES3D is evaluated using the metric Time DOF × RHS, (2) which measures the cost of evaluating the right-hand s… view at source ↗
Figure 2
Figure 2. Kinetic energy dissipation rate for standard (non-split) formulations with Gauss nodes. Both [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Effect of central split forms on dissipation and spectral behavior. The standard and Morinishi [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Effect of the Vreman SGS model on the Chandrasekhar split form. The model does not improve [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Comparison between the baseline central scheme and the same scheme augmented with a Vreman [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Effect of Riemann solver choice on dissipation and spectral behavior using the Chandrasekhar split [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Effect of combining Riemann solvers with a weak Vreman SGS model ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Inviscid TGV: Hybrid iLES–SGS configurations combining Riemann solvers with a weak Vreman [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Inviscid TGV: Effect of the Vreman SGS constant on dissipation and spectral behavior for hybrid [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Comparison between the baseline Roe scheme and the same scheme augmented with a Vreman [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems

    math.NA 2026-07 conditional novelty 4.0 of 10

    HORSES3D was GPU-ported with OpenACC, achieving near-ideal scaling above ~16–20k elements per GPU and running a 2,048-GPU, 10.7B-DOF High-Lift Common Research Model simulation.

Reference graph

Works this paper leans on

77 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods, Springer NewYork, 2008.URL:https://doi.org/10.1007/978-0-387-72067-8.doi:10.1007/ 978-0-387-72067-8

    work page doi:10.1007/978-0-387-72067-8.doi:10.1007/ 2008

  2. [2]

    Moura, S

    R. Moura, S. Sherwin, J. Peiro, Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: Insights into spectral van- ishing viscosity, Journal of Computational Physics 307 (2016) 401–422

    2016

  3. [3]

    Sherwin, Dispersion analysis of the continuous and discontinuous Galerkin formula- tions., in: in International Symposium on Discontinuous Galerkin Methods, Springer, 1999, pp

    S. Sherwin, Dispersion analysis of the continuous and discontinuous Galerkin formula- tions., in: in International Symposium on Discontinuous Galerkin Methods, Springer, 1999, pp. 425–431

    1999

  4. [4]

    Manzanero, G

    J. Manzanero, G. Rubio, E. Ferrer, E. Valero, Dispersion-dissipation analysis for ad- vection problems with nonconstant coefficients: Applications to discontinuous Galerkin formulations, SIAM Journal on Scientific Computing 40 (2018) A747–A768

    2018

  5. [5]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Berlin Heidelberg, 1988. URL:https://doi.org/10.1007/ 978-3-642-84108-8. doi:10.1007/978-3-642-84108-8

    doi:10.1007/978-3-642-84108-8 1988

  6. [6]

    T. A. Zang, On the rotation and skew-symmetric forms for incompressible flow simula- tions, Applied Numerical Mathematics 7 (1991) 27–40

    1991

  7. [7]

    Blaisdell, E

    G. Blaisdell, E. Spyropoulos, J. Qin, The effect of the formulation of nonlinear terms on aliasing errors in spectral methods, Applied Numerical Mathematics 21 (1996) 207–219

    1996

  8. [8]

    E. Ferrer, An interior penalty stabilised incompressible discontinuous Galerkin Fourier solver for implicit large eddy simulations, Journal of Computational Physics 348 (2017) 754 – 775. 24

    2017

Show all 77 references
  1. [9]

    R. M. Kirby, S. J. Sherwin, Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations, Journal of Engineering Mathematics 56 (2006) 273–288

    2006

  2. [10]

    R. M. Kirby, G. E. Karniadakis, De-aliasing on non-uniform grids: algorithms and applications, Journal of Computational Physics 191 (2003) 249–264

    2003

  3. [11]

    Mengaldo, D

    G. Mengaldo, D. D. Grazia, D. Moxey, P. Vincent, S. Sherwin, Dealiasing techniques for high-order spectral element methods on regular and irregular grids, Journal of Computational Physics 299 (2015) 56–81

    2015

  4. [12]

    H.Blackburn, S.Schmidt, Spectralelementfilteringtechniquesforlargeeddysimulation with dynamic estimation, Journal of Computational Physics 186 (2003) 610 – 629

    2003

  5. [13]

    T. C. Fisher, M. H. Carpenter, High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains, Journal of Computational Physics 252 (2013) 518–557

    2013

  6. [14]

    M. H. Carpenter, T. C. Fisher, E. J. Nielsen, S. H. Frankel, Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous interfaces, SIAM Journal on Scientific Computing 36 (2014) B835–B867

    2014

  7. [15]

    Schwarz, D

    A. Schwarz, D. Kempf, J. Keim, P. Kopper, C. Rohde, A. Beck, Comparison of entropy stable collocation high-order DG methods for compressible turbulent flows, Computers & Fluids 303 (2025) 106874

    2025

  8. [16]

    Manzanero, G

    J. Manzanero, G. Rubio, D. A. Kopriva, E. Ferrer, E. Valero, An entropy–stable discon- tinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility, Journal of Computational Physics 408 (2020) 109241

    2020

  9. [17]

    Manzanero, G

    J. Manzanero, G. Rubio, D. A. Kopriva, E. Ferrer, E. Valero, A free–energy stable nodal discontinuous Galerkin approximation with summation–by–parts property for the Cahn–Hilliard equation, Journal of Computational Physics 403 (2020) 109072

    2020

  10. [18]

    Lodares, J

    D. Lodares, J. Manzanero, E. Ferrer, E. Valero, An entropy–stable discontinuous Galerkin approximation of the Spalart–Allmaras turbulence model for the compress- ible Reynolds Averaged Navier–Stokes equations, Journal of Computational Physics 455 (2022) 110998. 25

    2022

  11. [19]

    Ntoukas, J

    G. Ntoukas, J. Manzanero, G. Rubio, E. Valero, E. Ferrer, A free–energy stable p– adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation, Journal of Com- putational Physics 442 (2021) 110409

    2021

  12. [20]

    G.Ntoukas, J.Manzanero, G.Rubio, E.Valero, E.Ferrer, Anentropy–stablep–adaptive nodal discontinuous Galerkin for the coupled Navier–Stokes/Cahn–Hilliard system, Journal of Computational Physics 458 (2022) 111093

    2022

  13. [21]

    A. R. Winters, R. C. Moura, G. Mengaldo, G. J. Gassner, S. Walch, J. Peiro, S. J. Sherwin, A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations, Journal of Computa- tional Physics 372 (2018) 1–21

    2018

  14. [22]

    Smagorinsky, General circulation experiments with the primitive equations: I

    J. Smagorinsky, General circulation experiments with the primitive equations: I. the basic experiment, Monthly weather review 91 (1963) 99–164

    1963

  15. [23]

    D. K. Lilly, On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems, Monthly Weather Review 93 (1965) 11–25

    1965

  16. [24]

    Nicoud, F

    F. Nicoud, F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow, turbulence and Combustion 62 (1999) 183–200

    1999

  17. [25]

    Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications, Physics of fluids 16 (2004) 3670–3681

    A. Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications, Physics of fluids 16 (2004) 3670–3681

    2004

  18. [26]

    D. J. Garmann, M. R. Visbal, P. D. Orkwis, Comparative study of implicit and subgrid- scale model large-eddy simulation techniques for low-Reynolds number airfoil applica- tions, International Journal for Numerical Methods in Fluids 71 (2013) 1546–1565

    2013

  19. [27]

    Y. Li, Z. Wang, A priori and a posteriori evaluations of sub-grid scale models for the Burgers’ equation, Computers & Fluids 139 (2016) 92–104

    2016

  20. [28]

    Z. Duan, Z. Wang, Calibrating sub-grid scale models for high-order wall-modeled large eddy simulation, Advances in Aerodynamics 6 (2024) 5

    2024

  21. [29]

    Chatterjee, Y

    T. Chatterjee, Y. T. Peet, Effect of artificial length scales in large eddy simulation of a neutral atmospheric boundary layer flow: A simple solution to log-layer mismatch, Physics of Fluids 29 (2017) 075105. 26

    2017

  22. [30]

    Mukha, P

    T. Mukha, P. Schlatter, Wall-modeled large-eddy simulation based on spectral-element discretization, arXiv preprint arXiv:2404.05378 (2024)

    2024 arXiv

  23. [31]

    Kumar, O

    V. Kumar, O. Lehmkuhl, A. Tomboulides, P. Fischer, M. Min, Turbulence Modeling with Nek5000/RS, SOD2D and Alya, Technical Report, Argonne National Laboratory (ANL), Argonne, IL (United States), 2023

    2023

  24. [32]

    Reddy, Y

    S. Reddy, Y. Tissaoui, Comparison of sub-grid scale models for large-eddy simulation using a high-order spectral element approximation of the compressible Navier-Stokes equations at low Mach number, Technical Report, 2021

    2021

  25. [33]

    Ntoukas, G

    G. Ntoukas, G. Rubio, O. A. Marino, A. Liosi, F. Bottone, J. Hoessler, E. Ferrer, A comparative study of explicit and implicit large eddy simulations using a high-order discontinuous Galerkin solver: Application to a Formula 1 front wing, Results in Engi- neering 25 (2025) 104425

    2025

  26. [34]

    A. Beck, M. Kurz, Toward discretization-consistent closure schemes for large eddy simulation using reinforcement learning, Physics of Fluids 35 (2023) 125122

    2023

  27. [35]

    M. Kurz, P. Offenhäuser, A. Beck, Deep reinforcement learning for turbulence modeling in large eddy simulations, International Journal of Heat and Fluid Flow 99 (2023) 109094

    2023

  28. [36]

    N. Fehn, W. A. Wall, M. Kronbichler, Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows, International Journal for Numerical Methods in Fluids 88 (2018) 32–54

    2018

  29. [37]

    Gasparino, F

    L. Gasparino, F. Spiga, O. Lehmkuhl, SOD2D: A GPU-enabled spectral finite elements method for compressible scale-resolving simulations, Computer Physics Communica- tions 297 (2024) 109067

    2024

  30. [38]

    Accessed: 2025-09-30

    HORSES3D: A high-order discontinuous Galerkin solver for flow simulations and multi- physics applications,https://github.com/loganoz/horses3d-gpu, 2025. Accessed: 2025-09-30

    2025

  31. [39]

    Kopriva, Implementingspectralmethodsforpartialdifferentialequations, Springer Netherlands, 2009

    D.A. Kopriva, Implementingspectralmethodsforpartialdifferentialequations, Springer Netherlands, 2009. URL:http://dx.doi.org/10.1007/978-90-481-2261-5. doi:10. 1007/978-90-481-2261-5. 27

    2009 doi

  32. [40]

    Ferrer, G

    E. Ferrer, G. Rubio, G. Ntoukas, W. Laskowski, O. A. Mariño, S. Colombo, A. Mateo- Gabín, H. Marbona, F. M. de Lara, D. Huergo, et al., HORSES3D: A high-order discon- tinuous Galerkin solver for flow simulations and multi-physics applications, Computer Physics Communications 2...

    2023

  33. [41]

    Gassner, D

    G. Gassner, D. A. Kopriva, A comparison of the dispersion and dissipation errors of Gauss and Gauss–Lobatto discontinuous Galerkin spectral element methods, SIAM Journal on Scientific Computing 33 (2011) 2560–2579

    2011

  34. [42]

    Manzanero, A

    J. Manzanero, A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, The Bassi Rebay 1 scheme is a special case of the symmetric interior penalty formulation for discontinuous Galerkin discretisationswithGauss–Lobattopoints, JournalofComputationalPhysics363(2018) 1–10

    2018

  35. [44]

    Arnold, F

    D. Arnold, F. Brezzi, B. Cockburn, L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal of Numerical Analysis 39 (2001) 1749– 1779

    2001

  36. [45]

    G. J. Gassner, A. D. Beck, On the accuracy of high-order discretizations for under- resolved turbulence simulations, Theoretical and Computational Fluid Dynamics 27 (2012) 221–237

    2012

  37. [46]

    Kopriva and G.J

    D.A. Kopriva and G.J. Gassner, An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems, SIAM Journal on Scientific Computing 36 (2014) A2076–A2099

    2014

  38. [47]

    G. J. Gassner, A. R. Winters, D. A. Kopriva, Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations, Jour- nal of Computational Physics 327 (2016) 39–66

    2016

  39. [48]

    Manzanero, G

    J. Manzanero, G. Rubio, E. Ferrer, E. Valero, D. A. Kopriva, Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods, J. Sci. Comput. 75 (2018) 1262–1281

    2018

  40. [49]

    Mateo-Gabín, A

    A. Mateo-Gabín, A. M. Rueda-Ramírez, E. Valero, G. Rubio, A flux-differencing for- mulation with Gauss nodes, Journal of Computational Physics 489 (2023) 112298. 28

    2023

  41. [50]

    D. A. Kopriva, G. J. Gassner, An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems, SIAM Journal on Scientific Computing 36 (2014) A2076–A2099

    2014

  42. [51]

    G. J. Gassner, A. R. Winters, F. J. Hindenlang, D. A. Kopriva, The BR1 scheme is stable for the compressible Navier–Stokes equations, Journal of Scientific Computing 77 (2018) 154–200

    2018

  43. [52]

    Manzanero, G

    J. Manzanero, G. Rubio, D. A. Kopriva, E. Ferrer, E. Valero, Entropy–stable discontin- uous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system, Journal of Computational Physics (2020) 109363

    2020

  44. [53]

    Chen, C.-W

    T. Chen, C.-W. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, Journal of Computational Physics 345 (2017) 427–461

    2017

  45. [54]

    G. J. Gassner, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM Journal on Scientific Computing 35 (2013) A1233–A1253

    2013

  46. [55]

    A. R. Winters, D. A. Kopriva, G. J. Gassner, F. Hindenlang, Construction of modern robust nodal discontinuous Galerkin spectral element methods for the compressible Navier–Stokes equations, in: Efficient High-Order Discretizations for Computational Fluid Dynamics, Springer, 20...

    2021

  47. [56]

    Chen, C.-W

    T. Chen, C.-W. Shu, Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes, CSIAM Trans. Appl. Math. 1 (2020) 1–52

    2020

  48. [57]

    Y. Morinishi, Skew-symmetric form of convective terms and fully conservative finite dif- ference schemes for variable density low-mach number flows, Journal of Computational Physics 229 (2010) 276–300

    2010

  49. [58]

    Ducros, F

    F. Ducros, F. Laporte, T. Soulères, V. Guinot, P. Moinat, B. Caruelle, High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: Application to compressible flows, Journal of Computational Physics 161 (2000) 114–139

    2000

  50. [59]

    C. A. Kennedy, A. Gruber, Reduced aliasing formulations of the convective terms within the Navier–Stokes equations for a compressible fluid, Journal of Computational Physics 227 (2008) 1676–1700. 29

    2008

  51. [60]

    Pirozzoli, Generalized conservative approximations of split convective derivative operators, Journal of Computational Physics 229 (2010) 7180–7190

    S. Pirozzoli, Generalized conservative approximations of split convective derivative operators, Journal of Computational Physics 229 (2010) 7180–7190

    2010

  52. [61]

    Ismail, P

    F. Ismail, P. L. Roe, Affordable, entropy-consistent euler flux functions II: Entropy production at shocks, Journal of Computational Physics 228 (2009) 5410–5436

    2009

  53. [62]

    P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Communications in Computational Physics 14 (2013) 1252–1286

    2013

  54. [63]

    Z. Wang, R. Fidkowski, K.and Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hartmann, K. Hillewaert, H. Huynh, N. Kroll, G. May, P. Persson, B. van Leer, M. Visbal, High-order CFD methods: current status and perspective, International Journal for Numerical Methods in...

    2013

  55. [64]

    A. Beck, T. Bolemann, D. Flad, H. Frank, G. Gassner, F. Hindenlang, C. Munz, High- order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations, International Journal for Numerical Methods in Fluids 76 (2014) 522–548

    2014

  56. [65]

    D. Flad, G. Gassner, On the use of kinetic energy preserving DG-schemes for large eddy simulation, Journal of Computational Physics 350 (2017) 782 – 795

    2017

  57. [66]

    Manzanero, E

    J. Manzanero, E. Ferrer, G. Rubio, E. Valero, On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods, arXiv:1805.10519 (2018)

    2018 arXiv

  58. [67]

    E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction, Springer Science & Business Media, 2013

    2013

  59. [68]

    Oßwald, A

    K. Oßwald, A. Siegmund, P. Birken, V. Hannemann, A. Meister, L2Roe: a low dissipa- tion version of roe’s approximate riemann solver for low mach numbers, International Journal for Numerical Methods in Fluids 81 (2016) 71–86

    2016

  60. [69]

    K. O. Friedrichs, P. D. Lax, Systems of conservation equations with a convex extension, Proceedings of the National Academy of Sciences 68 (1971) 1686–1688

    1971

  61. [70]

    Williamson, Low-storage Runge-Kutta schemes, Journal of Computational Physics 35 (1980) 48–56

    J. Williamson, Low-storage Runge-Kutta schemes, Journal of Computational Physics 35 (1980) 48–56. 30

    1980

  62. [71]

    N. Fehn, M. Kronbichler, P. Munch, W. A. Wall, Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid taylor–green problem, Journal of Fluid Mechanics 932 (2022) A40

    2022

  63. [72]

    J. R. Bull, A. Jameson, Simulation of the Taylor–Green vortex using high-order flux reconstruction schemes, AIAA Journal 53 (2015) 2750–2761

    2015

  64. [73]

    Carton De Wiart, K

    C. Carton De Wiart, K. Hillewaert, M. Duponcheel, G. Winckelmans, Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number, International Journal for Numerical Methods in Fluids 74 (2014) 469–493

    2014

  65. [74]

    Mateo-Gabin, K

    A. Mateo-Gabin, K. Tlales, E. Valero, E. Ferrer, G. Rubio, An unsupervised machine- learning-based shock sensor: Application to high-order supersonic flow solvers, Expert Systems with Applications 270 (2025) 126352

    2025

  66. [75]

    Vázquez, G

    M. Vázquez, G. Houzeaux, S. Koric, A. Artigues, J. Aguado-Sierra, R. Arís, D. Mira, H. Calmet, F. Cucchietti, H. Owen, et al., Alya: Multiphysics engineering simulation toward exascale, Journal of computational science 14 (2016) 15–27

    2016

  67. [76]

    Manzanero, E

    J. Manzanero, E. Ferrer, G. Rubio, E. Valero, Design of a Smagorinsky spectral van- ishing viscosity turbulence model for discontinuous Galerkin methods, Computers & Fluids 200 (2020) 104440

    2020

  68. [77]

    R. C. Moura, G. Mengaldo, J. Peiró, S. J. Sherwin, An LES setting for DG-based implicit LES with insights on dissipation and robustness, in: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016, Springer, 2017, pp. 161– 173

    2016

  69. [78]

    R. M. Kirby, S. J. Sherwin, Stabilisation of spectral/hp element methods through spec- tral vanishing viscosity: Application to fluid mechanics modelling, Computer methods in applied mechanics and engineering 195 (2006) 3128–3144. 31

    2006

Pith tools

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