Pith. sign in

REVIEW 62 references

A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations

T0 review · reviewed 2026-07-30 · grok-4.5

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

Pith's one-line read A spectral vanishing viscosity term keeps a high-order Navier–Stokes splitting scheme stable at high Reynolds number without losing its design accuracy.

desk verdict Clean fix of a real high-Re failure in Huang–Shen splitting; theory is honest but does not explain the Re=10^4 runs. read the letter →

arxiv 2607.23720 v1 pith:LAY5JIMV submitted 2026-07-26 math.NA cs.NA

M Nader Alhomsi , Akram Moustafa , Mohammad Al-Saqqa , Jiahong Wu , Xiaoming Zheng This is my paper · ORCID
classification math.NAcs.NA MSC 65M1265M7076D0565M15
keywords spectralvanishingviscosityconsistentsplittingschemeBDF–IMEXtimediscretizationincompressibleNavier–StokeserrorestimateshighReynoldsnumberKelvin–HelmholtzinstabilityKovasznayflow
topics Navier-Stokes Regularity
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

Higher-order consistent splitting schemes for the incompressible Navier–Stokes equations are attractive because they fully decouple velocity and pressure into standard elliptic solves, yet their error constants blow up as viscosity goes to zero and the bare schemes can fail at large Reynolds number. This paper shows that adding a directional spectral vanishing viscosity (SVV) operator to the velocity update restores stability: the operator damps only the high, under-resolved modes, costs nothing asymptotically, and leaves the structure of the existing error analysis intact. The authors prove stability and optimal-order error estimates in which SVV supplies a viscosity-independent coercive control of those high modes, while three two-dimensional tests confirm that the stabilized scheme of orders 2–4 retains design accuracy, resolves thin boundary layers, and tracks reference diagnostics where the unstabilized scheme diverges or blows up. A sympathetic reader cares because the method makes rigorously analyzed high-order splitting usable for the high-Reynolds regimes that originally motivated it.

What carries the argument

The directional SVV operator S_N = −ε_N div(Q_N ∇) built from the Maday–Kaber–Tadmor kernel applied separately in each coordinate; it is diagonal in the simultaneous-diagonalization eigenbasis, positive-semidefinite, free on resolved modes, and supplies the ν-independent high-mode coercivity in the energy and error estimates.

What would settle it

Run the stabilized scheme and the bare scheme on the manufactured solution or Kelvin–Helmholtz problem at Re=10^4 with the same spectral resolution: if the stabilized run loses design order or still blows up while matching the reference diagnostics fails, the central claim is false.

Watch

Extended reading notes

Core claim

Augmenting the Huang–Shen BDF–IMEX consistent splitting scheme with a directional Maday–Kaber–Tadmor spectral vanishing viscosity operator yields a scheme that remains stable and optimally accurate at high Reynolds number: SVV contributes a viscosity-independent coercive term on the high modes in the energy identity, the design temporal orders k=2,3,4 are retained, and the bare scheme’s breakdown at Re=10^4 is eliminated in manufactured, Kovasznay, and Kelvin–Helmholtz tests.

Load-bearing premise

The analysis assumes a strong solution with high temporal regularity on a smooth domain, which is not guaranteed for the high-Reynolds flows that motivate the method.

Editorial extensions

If this is right

  • High-order fully decoupled BDF–IMEX splitting becomes a practical option for under-resolved high-Re spectral computations without changing the per-step cost.
  • The same directional SVV correction carries over unchanged to every order k=2,3,4 and to Fourier–cosine/sine as well as Legendre–Galerkin bases.
  • Error constants still carry inverse powers of viscosity; SVV controls only high modes, so low-mode convection absorption remains ν-tied.
  • Three standard 2-D benchmarks (manufactured solution, perturbed Kovasznay, Kelvin–Helmholtz) become reliable testbeds for the stabilized family.

Reading between the lines

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

  • A quantitative low/high-mode split of the trilinear term could remove the remaining ν^{-5} factor and close the gap between the proved bound and the observed robustness.
  • The same diagonal SVV correction should extend immediately to three space dimensions once a tensor eigenbasis is available.
  • Adaptive or defect-corrected choices of ε_N and the cut-off m_N could lift the mild accuracy floor seen for k=3,4 without sacrificing stability.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: energy/error analysis is a standard a-priori estimate with an added PSD operator; numerics are checked against external or manufactured references.

full rationale

The load-bearing claim is Theorem 3.1 (stability and error bound (17) for the SVV-stabilized scheme (9)+(3b)). Its proof is a classical energy estimate: take the inner product of the discrete momentum equation with −δt ΔC_k(·), invoke G-stability of the shifted BDF (Lemma 3.5, from Huang–Shen [30], non-overlapping authors), B_k–C_k coercivity (Lemma 3.6), the new SVV coercivity (Lemma 3.7, obtained mode-by-mode by multiplying the same algebraic inequality by bQ_ij ≥ 0), Young absorption of convection/Stokes-pressure with ν-tied weights, and discrete Gronwall. The SVV operator S_N = −ε_N Q_N Δ is an independently defined symmetric positive-semidefinite spectral multiplier (Definition 2.2, Maday–Kaber–Tadmor kernel applied directionally); it is not fitted to the target error nor defined from the quantity being bounded. D_svv is a consistency remainder controlled by the exact solution, not a fitted prediction. Numerical claims are falsifiable against a manufactured solution, the classical Kovasznay base flow, and the external Kelvin–Helmholtz integral diagnostics of Schroeder et al. [47]. Self-citations ([1], [2]) supply background comparisons only and are not used to force uniqueness or close the main estimate. The acknowledged gap that the ν^{-5} Gronwall factor makes the theorem non-informative at Re = 10^4 is a usefulness/correctness issue, not circularity: the derivation does not reduce to its inputs by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central claim rests on classical Navier–Stokes regularity, the G-stability and B_k–C_k coercivity lemmas already proved by Huang–Shen, standard Stokes-pressure and trilinear estimates, and two hand-chosen SVV parameters. No new physical entities are introduced; the SVV operator is a known stabilization device placed inside a new scheme.

free parameters (3)
  • C_svv (SVV amplitude) = 1 (production)
    Sets ε_N = C_svv / M. Chosen as O(1); production runs use C_svv = 1. Controls the accuracy floor D_svv and the strength of high-mode damping.
  • m_N (SVV cut-off) = ⌈√M⌉
    Mode threshold below which the kernel vanishes. Set to ⌈√M⌉ following Maday–Kaber–Tadmor scaling; not optimized per problem.
  • β_k (Taylor-shift parameters) = 3,6,9
    Inherited recommended values β_2=3, β_3=6, β_4=9 from Huang–Shen; they enter every multiplier and the G-stability matrix.
assumptions (6)
  • domain assumption Assumption 2.1: strong solution with high temporal regularity on a C³ domain (u ∈ L^∞(H²∩H0¹), ∂_t^k u ∈ L^∞(H²), etc.)
    Invoked at the start of Theorem 3.1; supplies all exact-solution norms that appear in the data factor R_k and the induction hypothesis.
  • standard math G-stability of the shifted BDF multipliers (Lemma 3.5, from Huang–Shen / Dahlquist)
    Converts the discrete time derivative into a telescoping G_k-norm of the gradient history.
  • standard math B_k–C_k coercivity with η_k = 0.71 (Lemma 3.6, from Huang–Shen)
    Supplies the viscous and SVV coercive pieces after the algebraic splitting B_k = η_k C_k + D_k + F_k.
  • standard math Stokes-pressure estimate (Lemma 3.3, Liu–Liu–Pego)
    Controls ∥∇p_s(v)∥ by (1/2+ε)∥Δv∥² + C∥∇v∥²; used in both stability and pressure-error steps.
  • standard math Trilinear estimates of Lemma 3.4 / (20) for H² ∩ H0¹ vector fields
    Bound the convective terms that produce the ν^{-3} and ν^{-5} powers after Young absorption.
  • domain assumption Directional Maday–Kaber–Tadmor kernel with ε_N ∼ 1/M, m_N ∼ √M yields a self-adjoint positive-semidefinite operator that vanishes on low modes
    Definition 2.2 and Lemma 2.3; classical SVV facts used to obtain the extra coercive term without destroying consistency on resolved modes.
invented entities (1)
  • SVV-stabilized Huang–Shen scheme (eq. 9) independent evidence
    purpose: Replace the bare velocity update by one that includes −ε_N Q_N Δ B_k(u^{n+1}) so that high modes are damped.
    The scheme itself is the object of study; it is a concrete discretization, not a new physical entity. Independent evidence is the numerical comparison against bare scheme and external references.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations." pith.science (2026). https://pith.science/paper/LAY5JIMV

@misc{pith2026260723720,
  author       = {Pith},
  title        = {Pith review of: A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAY5JIMV}},
  note         = {Machine review of arXiv:2607.23720}
}
read the original abstract

Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.

Figures

Figures reproduced from arXiv: 2607.23720 by the authors.

Figure 1
Figure 1. Example 2. Kovasznay flow uKov at Re = 104 . The red region is where the first component uKov,1 is negative. indicating that the spatial discretization error dominates in this setting. Therefore, below we report results for different spatial resolutions while fixing k = 2 and δt = 10−4 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. [a] shows ∥w(t)∥ at Re = 102 , 103 , 104 at N = 128, with and without SVV. The SVV results indicate that the perturbation decays exponentially and the long-time solution returns to the Kovasznay flow. These curves are insensitive to the resolution: the SVV results for N = 128, 256, 512, 1024 coincide at each Reynolds number, as shown for Re = 104 in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Example 2 perturbation vorticity fields ∇ × w at t = 10 for Re = 104 , N = 128 (left) and N = 1024 (right) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Example 2 perturbation vorticity ∇ × w near right boundary for Re = 104 . From left to right: N = 128, 256, 512, 1024 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: shows the profiles of the perturbation quantities w = (w1, w2), ∇ × w and p − pKov along y = 0 for Re = 104 . The boundary layer thickness, distance from the outflow wall to the peak of w2 along y = 0, converges to 1.3 × 10−2 under mesh refinement. This layer contains …
Figure 6
Figure 6. Figure 6: Example 2. Outflow layer along y = 0 at N = 256 for Re = 102 (green), 103 (blue) and 104 (red). Upper row: peak of |w2|, the whole-domain norm ∥w∥L2(Ω), and the wall derivative |∂xw2(1, 0)|, on a logarithmic scale. Lower row: the distance 1 − xpeak and the two ratios o…
Figure 7
Figure 7. Figure 7: Example 2 at Re = 104 and t = 1, comparing the spectral scheme with N = 1024 against the Galerkin–Newton solver with N = 512. The two left panels show the perturbation vorticity ∇ × w near the outflow wall for the two methods. The two right panels show w2 along y = 0 o…
Figure 8
Figure 8. Figure 8: shows the early evolution of the vorticity field ∇×u for the SVV-stabilized scheme with k = 4 and N = 512, comparing Re = 103 (top row) with Re = 104 (bottom row) on the common colour scale of [47]. At t = 0 the two rows are identical, since the initial shear layer is …
Figure 9
Figure 9. Figure 9: Kelvin–Helmholtz problem at the coarse resolution [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Kelvin–Helmholtz problem at N = 512 with the SVV-stabilized scheme: vorticity ∇ × u for Re = 103 (top row) and Re = 104 (bottom row). In each row the left three panels are t = 12 and the right three are t = 20, with k = 2, 3, 4 within each group. Common colour scale o…
Figure 11
Figure 11. Figure 11: Kelvin–Helmholtz integral diagnostics at [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 24 canonical work pages

  1. [1]

    Nader Alhomsi, Jiahong Wu, and Xiaoming Zheng

    M. Nader Alhomsi, Jiahong Wu, and Xiaoming Zheng. Viscosity in error upper bound for a consistent splitting scheme of the Navier–Stokes equations.Journal of Computational and Applied Mathematics, page 117965, 2026. ISSN 0377-0427. doi: 10.1016/j.cam.2026.117965. 28

    arXiv 2026

  2. [2]

    Nader Alhomsi, Jiahong Wu, and Xiaoming Zheng

    M. Nader Alhomsi, Jiahong Wu, and Xiaoming Zheng. A consistent-splitting generalized scalar auxiliary variable scheme for the perturbed Boussinesq system.https://arxiv. org/abs/2606.31152, 2026. Submitted for publication

    arXiv 2026

  3. [3]

    Andreassen, I

    Ø. Andreassen, I. Lie, and C. E. Wasberg. The spectral viscosity method applied to simulation of waves in a stratified atmosphere.J. Comput. Phys., 110(2):257–273, 1994. doi: 10.1006/jcph.1994.1023

    arXiv 1994

  4. [4]

    Brezzi and M

    F. Brezzi and M. Fortin.Mixed and Hybrid Finite Element Methods, volume 15 ofSpringer Series in Computational Mathematics. Springer-Verlag, New York, 1991. doi: 10.1007/ 978-1-4612-3172-1

    1991

  5. [5]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang.Spectral Methods: Fundamen- tals in Single Domains. Springer, Berlin, 2006

    2006

  6. [6]

    G.-Q. Chen, Q. Du, and E. Tadmor. Spectral viscosity approximations to multidimensional scalar conservation laws.Math. Comp., 61(204):629–643, 1993. doi: 10.2307/2153244

    work page doi:10.2307/2153244 1993

  7. [7]

    L. Chen, R. Pasquetti, and C. Xu. Efficient SVV stabilized triangular spectral element methods for incompressible flows of high Reynolds numbers.Adv. Aerodyn., 3:27, 2021. doi: 10.1186/s42774-021-00090-x

    work page doi:10.1186/s42774-021-00090-x 2021

  8. [8]

    A. J. Chorin. Numerical solution of the Navier–Stokes equations.Math. Comp., 22(104): 745–762, 1968. doi: 10.1090/S0025-5718-1968-0242392-2

    doi:10.1090/s0025-5718-1968-0242392-2 1968

Show all 62 references
  1. [9]

    Dahlquist

    G. Dahlquist. G-stability is equivalent to A-stability.BIT, 18(4):384–401, 1978. doi: 10.1007/BF01932018

    1978 doi

  2. [10]

    de Frutos, B

    J. de Frutos, B. Garc ´ ıa-Archilla, V. John, and J. Novo. Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements.J. Sci. Comput., 66(3): 991–1024, 2016. doi: 10.1007/s10915-015-0052-1

    2016 doi

  3. [11]

    Deuflhard

    P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27(4):505–535, 1985

    1985

  4. [12]

    E and J.-G

    W. E and J.-G. Liu. Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal., 32(4):1017–1057, 1995. doi: 10.1137/0732047

    1995 doi

  5. [13]

    E and J.-G

    W. E and J.-G. Liu. Gauge method for viscous incompressible flows.Commun. Math. Sci., 1(2):317–332, 2003. doi: 10.4310/cms.2003.v1.n2.a6

    2003 doi

  6. [14]

    H. C. Elman, D. J. Silvester, and A. J. Wathen.Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford Univ. Press, Oxford, 2nd edition, 2014. doi: 10.1093/acprof:oso/9780199678792.001.0001

    2014

  7. [15]

    Garc ´ ıa-Archilla, V

    B. Garc ´ ıa-Archilla, V. John, and J. Novo. On the convergence order of the finite element er- ror in the kinetic energy for high Reynolds number incompressible flows.Comput. Methods Appl. Mech. Engrg., 385:114032, 2021. doi: 10.1016/j.cma.2021.114032

    2021

  8. [16]

    Garc ´ ıa-Archilla, V

    B. Garc ´ ıa-Archilla, V. John, and J. Novo. Error analysis of BDF schemes for the evolu- tionary incompressible Navier–Stokes equations.arXiv preprint arXiv:2506.16917, 2025. doi: 10.48550/arXiv.2506.16917

    2025 doi arXiv

  9. [17]

    Girault and P.-A

    V. Girault and P.-A. Raviart.Finite Element Approximation of the Navier–Stokes Equa- tions, volume 749 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1979. 29

    1979

  10. [18]

    Guermond and A

    J.-L. Guermond and A. J. Salgado. Error analysis of a fractional time-stepping technique for incompressible flows with variable density.SIAM J. Numer. Anal., 49(3):917–944, 2011. doi: 10.1137/090768758

    2011 doi

  11. [19]

    J. L. Guermond and J. Shen. A new class of truly consistent splitting schemes for incom- pressible flows.J. Comput. Phys., 192(1):262–276, 2003. doi: 10.1016/j.jcp.2003.07.009

    2003 doi

  12. [20]

    J. L. Guermond and J. Shen. Velocity-correction projection methods for incompressible flows.SIAM J. Numer. Anal., 41(1):112–134, 2003. doi: 10.1137/S0036142901395400

    2003 doi

  13. [21]

    J. L. Guermond and J. Shen. On the error estimates for the rotational pressure- correction projection methods.Math. Comp., 73(248):1719–1737, 2004. doi: 10.1090/ S0025-5718-03-01621-1

    2004

  14. [22]

    J. L. Guermond, P. Minev, and J. Shen. An overview of projection methods for incom- pressible flows.Comput. Methods Appl. Mech. Engrg., 195(44-47):6011–6045, 2006. doi: 10.1016/j.cma.2005.10.010

    2006 doi

  15. [23]

    Guermond, P

    J.-L. Guermond, P. D. Minev, and A. J. Salgado. Convergence analysis of a class of massively parallel direction splitting algorithms for the Navier–Stokes equations in simple domains.Math. Comp., 81(280):1951–1977, 2012. doi: 10.1090/S0025-5718-2012-02588-9

    1951 doi

  16. [24]

    Guo, H.-P

    B.-Y. Guo, H.-P. Ma, and E. Tadmor. Spectral vanishing viscosity method for nonlin- ear conservation laws.SIAM J. Numer. Anal., 39(4):1254–1268, 2001. doi: 10.1137/ S0036142900374233

    2001

  17. [25]

    Hairer and G

    E. Hairer and G. Wanner.Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems, volume 14 ofSpringer Series in Computational Mathematics. Springer- Verlag, Berlin, 2nd edition, 1996

    1996

  18. [26]

    Hairer, S

    E. Hairer, S. P. Nørsett, and G. Wanner.Solving Ordinary Differential Equations I: Nonstiff Problems, volume 8 ofSpringer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition, 1993

    1993

  19. [27]

    R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, Cambridge, 2nd edition, 2013

    2013

  20. [28]

    Huang and J

    F. Huang and J. Shen. Stability and error analysis of a second-order consistent splitting scheme for the Navier–Stokes equations.SIAM J. Numer. Anal., 61(5):2408–2433, 2023. doi: 10.1137/23M1556022

    2023 doi

  21. [29]

    Huang and J

    F. Huang and J. Shen. On a new class of BDF and IMEX schemes for parabolic type equations.SIAM J. Numer. Anal., 62(4):1609–1637, 2024. doi: 10.1137/23M1612986

    2024 doi

  22. [30]

    Huang and J

    F. Huang and J. Shen. Stability and error analysis of a new class of higher-order consistent splitting schemes for the navier-stokes equations.Math. Comp., 2025. doi: https://doi. org/10.1090/mcom/4132

    2025 doi

  23. [31]

    Johnston and J.-G

    H. Johnston and J.-G. Liu. Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term.J. Comput. Phys., 199(1):221–259, 2004. doi: 10.1016/j.jcp.2004.02.009

    2004 doi

  24. [32]

    Karamanos and G

    G.-S. Karamanos and G. E. Karniadakis. A spectral vanishing viscosity method for large- eddy simulations.J. Comput. Phys., 163(1):22–50, 2000. doi: 10.1006/jcph.2000.6552

  25. [33]

    G. E. Karniadakis, M. Israeli, and S. A. Orszag. High-order splitting methods for the incompressible Navier–Stokes equations.J. Comput. Phys., 97(2):414–443, 1991. doi: 10.1016/0021-9991(91)90007-8. 30

    1991 doi

  26. [34]

    R. M. Kirby and S. J. Sherwin. Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling.Comput. Methods Appl. Mech. Engrg., 195(23-24):3128–3144, 2006. doi: 10.1016/j.cma.2004.09.019

    2006 doi

  27. [35]

    L. I. G. Kovasznay. Laminar flow behind a two-dimensional grid.Math. Proc. Cambridge Philos. Soc., 44(1):58–62, 1948. doi: 10.1017/S0305004100023999

    1948 doi

  28. [36]

    J.-G. Liu, J. Liu, and R. L. Pego. Stable and accurate pressure approximation for unsteady incompressible viscous flow.Comm. Pure Appl. Math., 60(10):1443–1487, 2007. doi: 10. 1002/cpa.20178

    2007

  29. [37]

    Maday, S

    Y. Maday, S. M. Ould Kaber, and E. Tadmor. Legendre pseudospectral viscosity method for nonlinear conservation laws.SIAM J. Numer. Anal., 30(2):321–342, 1993. doi: 10. 1137/0730016

    1993

  30. [38]

    Ahmed body

    M. Minguez, R. Pasquetti, and E. Serre. High-order large-eddy simulation of flow over the “Ahmed body” car model.Phys. Fluids, 20(9):095101, 2008. doi: 10.1063/1.2952595

    2008 doi

  31. [39]

    R. C. Moura, S. J. Sherwin, and J. Peir´ o. Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection–diffusion problems: Insights into spectral vanishing viscosity.J. Comput. Phys., 307:401–422, 2016. doi: 10.1016/j.jcp.2015.12.009

    2016 doi

  32. [40]

    R. H. Nochetto and J.-H. Pyo. Error estimates for semi-discrete gauge methods for the Navier–Stokes equations.Math. Comp., 74(250):521–542, 2005. doi: 10.1090/ S0025-5718-04-01687-4

    2005

  33. [41]

    M. A. Olshanskii and L. G. Rebholz. Longer time accuracy for incompressible Navier– Stokes simulations with the EMAC formulation.Computer Methods in Applied Mechanics and Engineering, 372:113369, 2020. doi: 10.1016/j.cma.2020.113369

    2020

  34. [42]

    S. A. Orszag, M. Israeli, and M. O. Deville. Boundary conditions for incompressible flows. J. Sci. Comput., 1(1):75–111, 1986

    1986

  35. [43]

    Pasquetti

    R. Pasquetti. Spectral vanishing viscosity method for large-eddy simulation of turbulent flows.J. Sci. Comput., 27(1-3):365–375, 2006. doi: 10.1007/s10915-005-9029-9

    2006 doi

  36. [44]

    Pasquetti and C

    R. Pasquetti and C. J. Xu. High-order algorithms for large-eddy simulation of incompress- ible flows.J. Sci. Comput., 17(1-4):273–284, 2002. doi: 10.1023/A:1015113017729

    2002 doi

  37. [45]

    Prohl.Projection and Quasi-compressibility Methods for Solving the Incompressible Navier–Stokes Equations

    A. Prohl.Projection and Quasi-compressibility Methods for Solving the Incompressible Navier–Stokes Equations. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart,

  38. [46]

    Z. J. Rong and C. J. Xu. Spectral vanishing viscosity for large-eddy simulations by spectral element methods.Chinese J. Theor. Appl. Mech., 41(6):155–161, 2009

    2009

  39. [47]

    P. W. Schroeder, V. John, P. L. Lederer, C. Lehrenfeld, G. Lube, and J. Sch¨ oberl. On ref- erence solutions and the sensitivity of the 2D Kelvin–Helmholtz instability problem.Com- puters & Mathematics with Applications, 77(4):1010–1028, 2019. doi: 10.1016/j.camwa. 2018.10.030

    2019 doi

  40. [48]

    S´ ev´ erac and E

    ´E. S´ ev´ erac and E. Serre. A spectral vanishing viscosity LES model for the simulation of turbulent flows within rotating cavities.J. Comput. Phys., 226(2):1234–1255, 2007. doi: 10.1016/j.jcp.2007.05.023

    2007 doi

  41. [49]

    J. Shen. On error estimates of projection methods for Navier–Stokes equations: first-order schemes.SIAM J. Numer. Anal., 29(1):57–77, 1992. doi: 10.1137/0729004. 31

    1992 doi

  42. [50]

    J. Shen. Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials.SIAM J. Sci. Comput., 15(6):1489–1505, 1994. doi: 10.1137/0915089

    1994 doi

  43. [51]

    J. Shen. Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. InMultiscale Modeling and Analysis for Materials Simulation, volume 22 ofLect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 147–195. World Sci. Publ., Hackensac...

    2012

  44. [52]

    Shen and X

    J. Shen and X. Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows.Discrete Contin. Dyn. Syst. Ser. B, 8(3):663– 676, 2007. doi: 10.3934/dcdsb.2007.8.663

    2007 doi

  45. [53]

    E. Tadmor. Convergence of spectral methods for nonlinear conservation laws.SIAM J. Numer. Anal., 26(1):30–44, 1989. doi: 10.1137/0726003

    1989 doi

  46. [54]

    E. Tadmor. Approximate solutions of nonlinear conservation laws.Lecture Notes in Math., 1697:1–149, 1998. doi: 10.1007/BFb0096351. Advanced Numerical Approximation of Non- linear Hyperbolic Equations (Cetraro, 1997), Springer

    1998 doi

  47. [55]

    Temam.Navier–Stokes Equations: Theory and Numerical Analysis

    R. Temam.Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam, 3rd edition, 1984

    1984

  48. [56]

    L. J. P. Timmermans, P. D. Minev, and F. N. van de Vosse. An approximate projection scheme for incompressible flow using spectral elements.Internat. J. Numer. Methods Fluids, 22(7):673–688, 1996

    1996

  49. [57]

    Wang and J.-G

    C. Wang and J.-G. Liu. Convergence of gauge method for incompressible flow.Math. Comp., 69(232):1385–1407, 2000. doi: 10.1090/S0025-5718-00-01248-5

    2000 doi

  50. [58]

    J. A. C. Weideman and L. N. Trefethen. The eigenvalues of second-order spectral dif- ferentiation matrices.SIAM Journal on Numerical Analysis, 25(6):1279–1298, 1988. doi: 10.1137/0725072

    1988 doi

  51. [59]

    K. Wu, F. Huang, and J. Shen. A new class of higher-order decoupled schemes for the incompressible Navier–Stokes equations and applications to rotating dynamics.J. Comput. Phys., 458:111097, 2022. doi: 10.1016/j.jcp.2022.111097

    2022

  52. [60]

    Xu and R

    C. Xu and R. Pasquetti. Stabilized spectral element computations of high Reynolds number incompressible flows.J. Comput. Phys., 196(2):680–704, 2004. doi: 10.1016/j.jcp.2003.11. 009

    2004 doi

  53. [61]

    C. J. Xu. Stabilization methods for spectral element computations of incompressible flows. J. Sci. Comput., 27(1-3):495–505, 2006. doi: 10.1007/s10915-005-9059-3. 32

    2006 doi

  54. [1997]

    doi: 10.1007/978-3-663-11171-9

    doi

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.