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Original-energy-dissipation-preserving methods for the incompressible Navier-Stokes equations

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A bilinear rewrite of the Navier-Stokes equations makes exact energy conservation compatible with linear, cheap time stepping.

desk verdict A genuinely new linear reformulation of the convective term that preserves the original energy law; the unique-solvability worry is answerable and the main remaining issues are presentational. read the letter →

arxiv 2506.07141 v2 pith:HASOOKA3 submitted 2025-06-08 math.NA cs.NA

Zihan Weng , Qi Hong , Chunwu Wang , Yuezheng Gong This is my paper
classification math.NAcs.NA MSC 65M0665M1276D05
keywords robustreformulationincompressibleNavier-Stokesequationsenergydissipationlawstructure-preservingalgorithmsstaggeredgridfinitedifferencesCrank-NicolsonschemebackwarddifferentiationformulageneralizedStokessystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the incompressible Navier-Stokes equations can be rewritten in an equivalent form whose convection term is both unchanged on the solution and harmless under energy tests: a constructed bilinear operator $B(u,v)$ satisfies $B(u,u)=u\cdot\nabla u$ and $(B(u,v),v)=0$. Starting from this robust reformulation, the authors build four linear time-stepping schemes (Crank-Nicolson and backward differentiation formulas, first and second order) that reproduce the original energy dissipation law $\frac{d}{dt}\frac12\|u\|^2=-\nu\|\nabla u\|^2$ at every step, with no auxiliary variables. A finite-difference discretization on staggered grids carries the same exact dissipation to the fully discrete level, and each time step requires only three generalized Stokes solves and one $2\times2$ linear system. If the construction works as claimed, it gives a practical route to long incompressible-flow simulations at large Reynolds numbers and coarse time steps without sacrificing the governing energy law.

What carries the argument

The central object is the bilinear operator $B(u,v)=(F(u),v)G(u)-(G(u),v)F(u)$ with $G(u)=u\cdot\nabla u/(F(u),u)$, and its two identities $B(u,u)=u\cdot\nabla u$ and $(B(u,v),v)=0$. The first identity makes the reformulated PDE equivalent to Navier-Stokes; the second kills the convection term whenever the energy balance is tested against the new unknown. Because $B$ is linear in its second argument, replacing $u\cdot\nabla u$ by $B(u,v)$ with a frozen first argument turns each step into a linear generalized Stokes system, while the orthogonality converts the discrete energy test into exactly the continuous dissipation law. Recombination of the three Stokes solutions is controlled by the two scalars $\alpha=(F(\cdot),u)$ and $\beta=(G(\cdot),u)$, which are determined by the $2\times2$ linear system (4.17).

What would settle it

Run the fully discrete CN2 scheme on the classical vortex-flow benchmark with periodic boundary conditions and $F(u)=u$, and compute the determinant of the matrix $A$ in (4.17) at every time step; if the determinant reaches zero (or becomes ill-conditioned) for some choice of $\tau$, the claimed decomposition and unique solvability fail at that step. Independently, compare both sides of the discrete energy identity (4.7) in floating-point arithmetic over a long run; a persistent difference beyond solver tolerance would contradict the claim that the original energy dissipation law is exactly preserved.

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Extended reading notes

Core claim

For divergence-free $u$, choose any non-degenerate function $F(u)$ and define $G(u)=(u\cdot\nabla u)/(F(u),u)$ and $B(u,v)=(F(u),v)G(u)-(G(u),v)F(u)$. Then, under homogeneous Dirichlet or periodic boundary conditions, $B(u,u)=u\cdot\nabla u$ and $(B(u,v),v)=0$, so the system $u_t-\nu\Delta u+B(u,u)+\nabla p=f$, $\nabla\cdot u=0$ is exactly the incompressible Navier-Stokes system. The paper's point is structural: because $B$ is linear in its second argument, freezing the first argument at a known state produces a linear Stokes-type problem, and the orthogonality $(B(u,v),v)=0$ makes the energy test yield exactly $\frac{d}{dt}\frac12\|u\|^2=-\nu\|\nabla u\|^2$. Four time-discrete schemes (first- and second-order linear CN and BDF) and their fully discrete staggered-grid versions are built on this identity; the paper proves that every one of them conserves the discrete version of the original energy dissipation law and has a unique solution, and it gives an implementation that needs only three generalized Stokes solves plus a $2\times2$ system per step.

Load-bearing premise

At each time step the implementation must solve a $2\times2$ linear system for two scalar coefficients, and the paper assumes, rather than proves, that this $2\times2$ matrix is invertible; if it were singular, the decomposition into three Stokes solves could not be formed and the unique-solvability claim would not apply.

Editorial extensions

If this is right

  • The CN1, CN2, BDF1, and BDF2 time-discrete schemes are all linear and preserve the original energy dissipation law exactly, without auxiliary variables.
  • Each time step costs three generalized Stokes solves and one $2\times2$ linear system; under periodic boundary conditions the Poisson and Helmholtz subproblems can be solved by FFT.
  • The fully discrete staggered-grid schemes inherit the same discrete energy law and are proved uniquely solvable, for both periodic and homogeneous Dirichlet boundary conditions.
  • Numerical experiments on vortex flows, cavity flows, and shear-layer instability confirm second-order convergence and long-time energy decay, with accuracy at coarse time steps strongly dependent on the choice of $F(u)$.

Reading between the lines

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

  • The same construction should transfer to any hydrodynamic or phase-field model whose nonlinearity is skew-symmetric under the $L^2$ inner product; this is an extrapolation the paper only hints at in its conclusion.
  • The monitored quantity $|(u^{n+1/2}\cdot\nabla u^{n+1/2},u^{n+1/2})_h|$ behaves like an a posteriori measure of how far the frozen linearization is from the true convection, so it could be used as an adaptive time-step controller; the paper leaves this as future work.
  • If the $2\times2$ coefficient matrix in (4.17) is ever singular or ill-conditioned, the three-Stokes decomposition would fail, so a fully unconditional implementation would need an adaptive choice of $F(u)$ or a fallback direct solve of the coupled linear system; this is a caution beyond the paper's theorems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper proposes a 'robust reformulation' of the incompressible Navier-Stokes equations, replacing the convective term u·∇u by a bilinear form B(u,v) that equals u·∇u when v=u and satisfies the skew-symmetry (B(u,v),v)=0. This allows the authors to construct linearly implicit Crank-Nicolson and BDF time discretizations, and fully discrete staggered-grid finite-difference schemes, that satisfy discrete analogues of the original energy dissipation law while requiring only three generalized Stokes solves and one 2x2 linear system per time step. The discrete energy identities are proved in Theorems 3.1--3.3, 4.1, and 4.3, and unique solvability of the fully discrete schemes is addressed in Theorems 4.2 and 4.4. Numerical experiments verify convergence, energy dissipation, and performance on standard benchmarks. The central algebraic construction is correct, but the paper omits a proof that the 2x2 system used in the efficient implementation is nonsingular, and the statement of the second-order schemes uses the same symbol for an extrapolated known velocity and the unknown midpoint/next-step velocity.

Significance. If the results are accepted, the paper contributes a simple and elegant framework for designing linearly implicit, unconditionally energy-stable schemes for the incompressible Navier-Stokes equations that preserve the original physical energy dissipation law, rather than a modified law with auxiliary variables. The computational cost—three generalized Stokes solves and a 2x2 system per step—is competitive with recent SAV and Lagrange-multiplier approaches, and the numerical experiments indicate improved accuracy at large time steps. The proof of the energy identities is direct and algebraic, and the fully discrete staggered-grid formulation is carefully constructed so that the key skew-symmetry persists. The lack of rigorous error estimates is a limitation, but it does not undermine the stated structure-preservation and solvability claims, which are the paper's main focus. The method is likely to be of interest to researchers in structure-preserving discretizations for incompressible flows.

major comments (1)
  1. [Section 4.2, Eq. (4.17); Section 4.3, Eq. (4.34)] The efficient implementation depends on the invertibility of the 2x2 coefficient matrix A in (4.17) and (4.34), but the paper never proves that this matrix is nonsingular. The uniqueness proofs in Theorems 4.2 and 4.4 establish unique solvability of the full coupled system via the homogeneous Stokes-type system, yet they do not imply that the decomposed 2x2 system has a solution; if A were singular, the proposed three-Stokes-solves-plus-2x2 implementation would fail even though the scheme itself is solvable. This is load-bearing for the paper's central efficiency claim. I recommend adding a short proof, e.g., letting S denote the symmetric positive definite operator (2/τ)I - νΔ_h on the divergence-free subspace and L the solution operator of the generalized Stokes problem, one can show det(A) = 1 - c^2 + ab with a=(F,LF)≥0, b=(G,LG)≥0, c=-(F,LG), and |c|≤√(ab) by Cauchy-Schwarz, so det(A)≥1. The same argument applies to the semi-discrete systems (3.10) and (3.23).
minor comments (4)
  1. [Schemes 3.2, 3.4, 4.2, 4.4] The symbol u^{n+1/2} (or U^{n+1/2}, u^{n+1}) is first defined as a second-order extrapolation of known values, but then used in the scheme equations for the unknown midpoint or next-step velocity in the convective term B(·,·) and in the divergence constraint. This makes the scheme statements ill-posed as written. Please use separate notations, such as \tilde{u}^{n+1/2} for the extrapolated argument and u^{n+1/2} for the unknown midpoint, and likewise for the BDF2 schemes.
  2. [General] The paper does not provide any error estimates for the proposed time-stepping or fully discrete schemes. The numerical convergence tests are convincing, but a statement that rigorous error analysis is deferred to future work would set expectations appropriately.
  3. [Abstract and Section 3.2] The abstract states that all four schemes 'exactly preserve the original energy dissipation law at each time step.' For the BDF1 and BDF2 schemes, the discrete energy laws (3.13) and (3.16) contain additional numerical dissipation terms, so they do not reproduce the continuous law exactly. Consider qualifying the statement to distinguish the CN schemes (which match the continuous form exactly) from the BDF schemes (which dissipate slightly more).
  4. [Throughout] There are several typographical errors: 'lemme' in Lemmas 2.1 and 4.1 should be 'lemma'; 'girds' in the key words should be 'grids'; 'effcient' in Section 4.3 should be 'efficient'; 'finial time' in Section 5.1 should be 'final time'; and 'unique solvable' in Section 4.3 should be 'uniquely solvable'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the energy-preserving property is an algebraic consequence of the defined operator B, and the empirical choice of F does not enter the energy theorems.

full rationale

The central derivation is self-contained. Section 2 defines G and B by (2.3)-(2.4); Lemma 2.1 proves B(u,u)=u·∇u and (B(u,v),v)=0 directly from the definition, so the robust reformulation (2.9) is exactly equivalent to (1.1), not a fitted prediction. Theorem 2.1 and the time-discrete Theorems 3.1-3.3 are direct inner-product identities: taking the inner product of each scheme with the appropriate velocity eliminates B by (2.6) and yields exactly -ν||∇u||^2 or its BDF modification. The fully discrete Theorems 4.1 and 4.3 use the same algebraic property of the staggered-grid B(U,V), so the claimed original energy dissipation law is proved from the definitions, not imported from data. Unique solvability (Theorems 4.2 and 4.4) is proved by showing the homogeneous linear system has only the zero solution; the invocation of Theorem 5.1 of Ref. [17] is a standard injectivity-to-existence step for finite-dimensional square Stokes-like systems and the paper supplies the needed injectivity argument, so the self-citation is independent support rather than a circular premise. The numerical choice F4(u) is empirical, but every energy theorem holds for any admissible F satisfying (2.2), so the choice cannot be a fitted input that forces the central result. The acknowledged presentation issue in Schemes 3.2 and 4.2, where the symbol for the midpoint velocity is reused for the extrapolated argument, is a notation defect and does not make the derivation circular.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

All energy results follow from the built-in identity (B(U,V),V)_h=0, so the derivation is self-contained. The unproved 2x2 invertibility is the main load-bearing assumption, and the helper function F is a hand-chosen stabilization rather than a fitted physical constant.

free parameters (2)
  • F(u) helper function = F4(u)=u^{-3} (with 1e-10 cutoff) recommended; F1=u used in several tests
    The reformulation works for any non-degenerate F, but coarse-step accuracy depends strongly on this hand-chosen function. The paper selects F4 after observing better energy curves on the Taylor-Green problem in Section 5.2.
  • Cutoff threshold in F3 and F4 = 1e-10
    Ad hoc threshold used to avoid division by zero in 1/u and 1/u^3; no analysis of its effect is given.
assumptions (4)
  • domain assumption For divergence-free u under Dirichlet or periodic conditions, (u·∇u,u)=0 (Eq 2.1).
    Used to prove B(u,u)=u·∇u and the energy law; standard for the stated boundary conditions.
  • standard math The discrete Laplacian Δ_h is negative semidefinite on the staggered-grid spaces.
    Used in Theorems 4.1 through 4.4 to conclude ν(Δ_h U,U)_h ≤0; follows from the periodic difference matrices but is not proved in the paper.
  • domain assumption The generalized Stokes subsystems (4.13) and (4.30) are uniquely solvable under mean-zero pressure.
    Invoked by reference to Theorem 5.1 in Ref [17]; standard for MAC-type discretizations.
  • ad hoc to paper The 2x2 system (4.17) is nonsingular.
    Required for existence via the alpha-beta decomposition; not proved in Theorem 4.2 or 4.4.

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Cite this review

Pith. "Pith review of Original-energy-dissipation-preserving methods for the incompressible Navier-Stokes equations." pith.science (2026). https://pith.science/paper/HASOOKA3

@misc{pith2026250607141,
  author       = {Pith},
  title        = {Pith review of: Original-energy-dissipation-preserving methods for the incompressible Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HASOOKA3}},
  note         = {Machine review of arXiv:2506.07141}
}
abstract

This paper introduces a robust reformulation of the incompressible Navier-Stokes equations, establishing a foundational framework for designing efficient, structure-preserving algorithms that strictly conserve the original energy dissipation law. By leveraging Crank-Nicolson schemes and backward differentiation formulas, we develop four first- and second-order time-discrete schemes. These schemes exactly preserve the original energy dissipation law at each time step, requiring only the solutions of three linear Stokes systems and one $2\times 2$ system of linear equations. Furthermore, the finite difference approximation on a staggered grid is employed for these time-discrete systems to derive fully discrete structure-preserving schemes. We rigorously prove that all proposed fully discrete methods both maintain the original energy dissipation law and admit unique solutions. Moreover, we present their efficient implementation. Extensive numerical experiments are carried out to verify the accuracy, efficacy, and advantageous performance of our newly developed methods.

Figures

Figures reproduced from arXiv: 2506.07141 by the authors.

Figure 1
Figure 1. In this schematic, Ωew corresponds to blue squares, Ωns to red triangles, and Ωc to black circles. Then we define the following discrete function spaces [17, 31] Vew = n U|U ={u(xi+ 1 2 ,yj)|(xi+ 1 2 ,yj)∈Ωew} o , Vns = n V|V ={v(xi ,yj+ 1 2 )|(xi ,yj+ 1 2 )∈Ωns} o , Vc =  P|P={p(xi ,yj)|(xi ,yj)∈Ωc} [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Errors of the velocity and pressure obtained by various schemes with [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Energy evolution for CN2 scheme with different forms of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Comparisons of the quantity |(u n+ 1 2 ·∇u n+ 1 2 ,u n+ 1 2 )h | between the CN2 scheme with various forms of F(u) [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Energy evolution for CN2 scheme with recommended [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Energy evolution for CN2 scheme with F4(u). Mesh size is h= 1 128 , τ= 1 64 for Re=1000, and h= 1 256 , τ= 1 256 for Re=10000 in the long-time simulation [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Streamline (left) and contour plots (right) of the velocity magnitude at the steady [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Velocity on x=0.5 and y=0.5 of the lid-driven cavity flow. For the 2D problem [35], we consider the computational domain Ω = [0,1] 2 , bounded by three stationary no-slip walls (at x = 0, x = 1, and y = 0) and a lid (at y = 1) moving with a tangential unit velocity. As…
Figure 9
Figure 9. Figure 9: Contour plots of the vorticity components on the midplanes at the steady state. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Steady-state velocity profiles along the cavity centerlines. [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Vorticity field at various times [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Energy evolution for CN2 scheme with h = 1 256 and 1 420 and the quantity |(u n+ 1 2 · ∇u n+ 1 2 ,u n+ 1 2 )h | . 6 Conclusion In this paper, we have proposed the robust reformulation for the incompressible NS equa￾tions to develop first- and second-order structure-pr…

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Works this paper leans on

38 extracted references · 35 canonical work pages

  1. [1]

    M. O. Deville, P . F. Fischer, E. H. Mund, High-order methods for incompressible fluid flow, Cam- bridge university press, 2002

    work page 2002

  2. [2]

    M. D. Gunzburger, Finite element methods for viscous incompressible flows: a guide to theory, practice, and algorithms, Academic Press, 1989

    work page 1989

  3. [3]

    Temam, Navier-Stokes equations: theory and numerical analysis, North Holland, 2016

    R. Temam, Navier-Stokes equations: theory and numerical analysis, North Holland, 2016

    work page 2016

  4. [4]

    Peyret, Spectral methods for incompressible viscous flow, Springer New York, 2002

    R. Peyret, Spectral methods for incompressible viscous flow, Springer New York, 2002

    work page 2002

  5. [5]

    A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput. 22 (1968) 745–762

    1968

  6. [6]

    Temam, Sur l’approximation de la solution des ´equations de Navier-Stokes par la m´ethode des pas fractionnaires (ii), Arch

    R. Temam, Sur l’approximation de la solution des ´equations de Navier-Stokes par la m´ethode des pas fractionnaires (ii), Arch. Ration. Mech. Anal. 33 (1969) 377–385

    work page 1969

  7. [7]

    J. Kim, P . Moin, Application of a fractional-step method to incompressible Navier-Stokes equa- tions, J. Comput. Phys. 59 (1985) 308–323

    work page 1985

  8. [8]

    Guermond, P

    J. Guermond, P . Minev, J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011–6045

    work page 2006

Show all 38 references
  1. [9]

    W. E, J. Liu, Projection Method I: Convergence and Numerical Boundary Layers, SIAM J. Numer. Anal. 32 (1995) 1017–1057

    1995

  2. [10]

    W. E, J. Liu, Gauge finite element method for incompressible flows, Int. J. Numer. Methods Fluids. 34 (2000) 701–710

    2000

  3. [11]

    Timmermans, P

    L. Timmermans, P . Minev, F. N. V . D. Vosse, An approximate projection scheme for incompressible flow using spectral elements, International journal for numerical methods in fluids 22 (1996) 673– 688

    1996

  4. [12]

    X. Li, J. Shen, Z. Liu, New SAV-pressure correction methods for the Navier-Stokes equations: sta- bility and error analysis, Math. Comput. 91 (2022) 141–167

    2022

  5. [13]

    Shen, On error estimates of projection methods for Navier-Stokes equations: First-order schemes, SIAM J

    J. Shen, On error estimates of projection methods for Navier-Stokes equations: First-order schemes, SIAM J. Numer. Anal. (1992) 57–77

    1992

  6. [14]

    Hairer, C

    E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-preserving Algo- rithms for Ordinary differential Equations, Springer Berlin, Heidelberg, 2006

    2006

  7. [15]

    J. Simo, F. Armero, Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg. 111 (1994) 111–154

    1994

  8. [16]

    Y. Gong, X. Liu, Q. Wang, Fully discretized energy stable schemes for hydrodynamic equations governing two-phase viscous fluid flows, J. Sci. Comput. 69 (2016) 921 – 945

    2016

  9. [17]

    Y. Gong, J. Zhao, Q. Wang, Second order fully discrete energy stable methods on staggered grids for hydrodynamic phase field models of binary viscous fluids, SIAM J. Sci. Comput. 40 (2018) B528–B553

    2018

  10. [18]

    Verstappen, A

    R. Verstappen, A. Veldman, Symmetry-preserving discretization of turbulent flow, J. Comput. Phys. 187 (2003) 343–368

    2003

  11. [19]

    J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353 (2018) 407–416

    2018

  12. [20]

    J. Shen, J. Xu, J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61 (2019) 474–506

    2019

  13. [21]

    L. Lin, Z. Yang, S. Dong, Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable, J. Comput. Phys. 388 (2019) 1–22

    2019

  14. [22]

    X. Li, J. Shen, Error analysis of the SAV-MAC scheme for the navier–stokes equations, SIAM J. Numer. Anal. 58 (2020) 2465–2491

    2020

  15. [23]

    K. Wu, F. Huang, J. Shen, A new class of higher-order decoupled schemes for the incompress- ible Navier-Stokes equations and applications to rotating dynamics, J. Comput. Phys. 458 (2022) 111097. 29

    2022

  16. [24]

    Obbadi, M

    A. Obbadi, M. El-Amrani, M. Seaid, D. Yakoubi, A stable second-order splitting method for incom- pressible Navier-Stokes equations using the scalar auxiliary variable approach, Comput. Methods Appl. Mech. Engrg. 437 (2025) 117801

    2025

  17. [25]

    Cheng, C

    Q. Cheng, C. Liu, J. Shen, A new Lagrange multiplier approach for gradient flows, Comput. Meth- ods Appl. Mech. Engrg. 367 (2020) 113070

    2020

  18. [26]

    J. Yang, Z. Tan, J. Kim, Original variables based energy-stable time-dependent auxiliary variable method for the incompressible Navier–Stokes equation, Comput. & Fluids 240 (2022) 105432

    2022

  19. [27]

    Doan, T.-T.-P

    C.-K. Doan, T.-T.-P . Hoang, L. Ju, R. Lan, Dynamically regularized lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations, J. Comput. Phys. 521 (2025) 113550

    2025

  20. [28]

    X. Yang, A novel fully decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier–Stokes equations coupled with mass-conserved Allen-Cahn phase- field model of two-phase incompressible flow, Internat. J. Numer. Methods Engrg. 122 (2021) ...

    2021

  21. [29]

    Yang, A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model, J

    X. Yang, A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model, J. Com- put. Phys. 432 (2021) 110015

    2021

  22. [30]

    Yang, A novel fully-decoupled, second-order and energy stable numerical scheme of the con- served Allen–Cahn type flow-coupled binary surfactant model, Comput

    X. Yang, A novel fully-decoupled, second-order and energy stable numerical scheme of the con- served Allen–Cahn type flow-coupled binary surfactant model, Comput. Methods Appl. Mech. Engrg. 373 (2021) 113502

    2021

  23. [31]

    Q. Hong, Y. Gong, J. Zhao, Thermodynamically consistent hydrodynamic phase-field computa- tional modeling for fluid-structure interaction with moving contact lines, J. Comput. Phys. 492 (2023) 112409

    2023

  24. [32]

    B. E. Griffith, An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner, J. Comput. Phys. 228 (2009) 7565–7595

    2009

  25. [33]

    L. Ju, J. Zhang, L. Zhu, Q. Du, Fast explicit integration factor methods for semilinear parabolic equations, J. Sci. Comput. 62 (2015) 431–455

    2015

  26. [34]

    Cheng, C

    K. Cheng, C. Wang, Long time stability of high order multistep numerical schemes for two- dimensional incompressible Navier–Stokes equations, SIAM J. Numer. Anal. 54 (2016) 3123–3144

    2016

  27. [35]

    T. A. AbdelMigid, K. M. Saqr, M. A. Kotb, A. A. Aboelfarag, Revisiting the lid-driven cavity flow problem: Review and new steady state benchmarking results using GPU accelerated code, Alex. Eng. J. 56 (2017) 123–135

    2017

  28. [36]

    Albensoeder, H

    S. Albensoeder, H. Kuhlmann, Accurate three-dimensional lid-driven cavity flow, J. Comput. Phys. 206 (2005) 536–558

    2005

  29. [37]

    Jiang, T

    B. Jiang, T. Lin, L. A. Povinelli, Large-scale computation of incompressible viscous flow by least- squares finite element method, Comput. Methods Appl. Mech. Engrg. 114 (1994) 213–231

    1994

  30. [38]

    P . W. Schroeder, V . John, P . L. Lederer, C. Lehrenfeld, G. Lube, J. Sch ¨oberl, On reference solu- tions and the sensitivity of the 2D Kelvin–Helmholtz instability problem, Comput. Math. Appl. 77 (2019) 1010–1028

    2019

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