Pith. sign in

REVIEW 2 major objections 5 minor 53 references

Generalized skew-gradient embedding for thermodynamically consistent systems

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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

Pith's one-line read The reversible part of a thermodynamically consistent system can be written many ways; least-squares picks a unique gauge that keeps the energy law and often decouples the numerics.

desk verdict Clean algebraic extension of the authors' SGE work: affine gauges, least-squares selection, and two solid discrete GENERIC schemes, with the usual compatibility caveat and no new numerics. read the letter →

arxiv 2607.09617 v2 pith:NBNTPJNO submitted 2026-07-10 math.NA cs.NA

Xuelong Gu , Qi Wang This is my paper · ORCID
classification math.NAcs.NA MSC 65M0665P1037N1076D05
keywords GENERICskew-gradientembeddingstructure-preservingdiscretizationPoissonstructureCahn–Hilliard–Navier–Stokesenergy-stableschemesgaugefreedom
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

Thermodynamically consistent continuum models split into reversible and irreversible pieces that must respect energy and entropy balance. Earlier work rewrote the reversible piece by embedding it in a single rank-two skew operator driven by the thermodynamic force, so that the whole system becomes a generalized gradient flow. This paper shows that embedding is not unique: every admissible two-form differs from a reference form by something that still annihilates the force, forming an affine family called generalized skew-gradient embeddings. For any positive-definite metric the unique minimum-norm representative is given by a simple weighted wedge product; the ordinary metric recovers the original embedding. The same formula supplies regularized gauges when the force vanishes, corrections of slightly non-neutral residuals, and gauges that preserve any prescribed list of invariants. A rank-two Jacobi identity then tells when the gauge is Poisson, allowing fully discrete GENERIC schemes. Applied to a marker-and-cell discretization of Navier–Stokes and to a regularized BDF2 scheme for Cahn–Hilliard–Navier–Stokes, the construction yields exact discrete energy laws and inexpensive decoupled solvers.

What carries the argument

The unified least-squares gauge (Theorem 3.1): among all two-forms whose contraction with the thermodynamic force recovers the reversible field, the weighted wedge ω_A,r is the unique minimizer of the A-Frobenius norm, and its action remains matrix-free.

What would settle it

On a uniform MAC grid for incompressible Navier–Stokes, compute the discrete Lie bracket of the constructed rank-two fields and check whether it remains inside their span; any nonzero component outside that span falsifies the claimed Poisson structure and the exact discrete free-energy law of the midpoint scheme.

Watch

Extended reading notes

Core claim

The admissible two-forms that embed the reversible field form an affine space. For any positive-definite metric A the unique minimum-Hilbert–Schmidt gauge is the weighted wedge (A X)^♭ ∧ r / ⟨A X, X⟩; the identity metric recovers the original skew-gradient embedding, while the same least-squares principle produces regularized, residual-correcting and invariant-preserving gauges that still obey the entropy or free-energy law.

Load-bearing premise

The spatial discretization must exactly inherit the continuous free-energy neutrality and the algebraic identities that make the discrete Jacobi identity and energy telescoping hold; if those identities are lost, the fully discrete structure and exact energy law collapse.

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.

Referee Report

2 major / 5 minor

Summary. The paper generalizes the authors’ earlier skew-gradient embedding (SGE) by characterizing all two-forms that embed the reversible GENERIC field L∇E (or L∇F) as an affine family of admissible gauges (GSGE). For any positive-definite metric A, weighted least squares selects a unique minimum-Hilbert–Schmidt representative that recovers SGE when A = I; the same construction yields regularized, residual-correcting, and invariant-preserving gauges that retain the entropy or free-energy law. A finite-dimensional rank-two Jacobi criterion is given for the isothermal case. The theory is applied to a compatible MAC discretization of incompressible Navier–Stokes, producing a semi-discrete rank-two Poisson structure that the implicit midpoint rule preserves fully discretely with an exact free-energy law, and to a regularized GSGE–BDF2 scheme for Cahn–Hilliard–Navier–Stokes that preserves mass, dissipates free energy unconditionally, and decouples through two scalar coefficients.

Significance. If the claims hold, the work supplies a clean algebraic toolkit for structure-preserving time discretizations of thermodynamically consistent multiphysics systems: an affine gauge space, a unique least-squares selection principle, regularization at vanishing force, and invariant-preserving projections, all retaining the discrete energy/entropy law while often decoupling subproblems. The finite-dimensional proofs (affine space, uniqueness of ω_A,r, regularized formula, Gram projection, rank-two Jacobi criterion) are self-contained and machine-checkable in principle. The MAC + midpoint construction yields a fully discrete isothermal GENERIC scheme with an exact free-energy identity, and the CHNS scheme inherits mass conservation and unconditional free-energy dissipation from the same gauge algebra. These are concrete, reusable contributions for geometric numerical integration of GENERIC and free-energy systems.

major comments (2)
  1. Theorems 4.2–4.3 and 4.6 rest on the hypothesis that the chosen spatial operators (MAC for NS; periodic summation-by-parts for CHNS) exactly inherit free-energy neutrality ⟨∇F, J⟩ = 0 and the algebraic identities needed for the discrete Lie bracket and energy telescoping. The manuscript states the needed identities (skewness of C_h, quadratic homogeneity of J_h, (J_h(u),u)_h = 0) but does not verify them for the concrete centered MAC averages of N_h and K_h, nor does it supply a short appendix or reference that does so for the precise stencil used. Without that verification the fully discrete GENERIC claim and exact energy law remain conditional on an unproved compatibility assumption.
  2. Section 4.2 and Remark 4.4 introduce a three-parameter regularized differential gauge (ℓ1, ℓ2, ℓ3) and assert second-order consistency when ℓ3 au^{2} = O( au^{2}) and d^{n+1} > 0, yet no truncation-error analysis or numerical confirmation is given. Because the paper explicitly declines new numerical experiments, a short consistency argument (or a pointer to the order analysis of the recovered SGE–SBDF2 limit) is needed to substantiate that the free parameters do not degrade the formal order of the BDF2 scheme.
minor comments (5)
  1. The abstract and introduction cite GuWangSGE2025 / arXiv:2509.18601 as the SGE baseline; ensure the published or final arXiv version is used consistently and that the present paper is self-contained for readers who have not seen that preprint.
  2. Notation for the musical maps ♭/♯ and the two wedge products (forms vs. maps) is introduced carefully in §2.1 but then used interchangeably; a one-line reminder when switching from ω to the map Y ∧ Z would help.
  3. Proposition 3.8 asserts local existence of a commuting Y with prescribed invariants; a brief remark on whether the construction extends globally on the affine phase space of CHNS (fixed mass, divergence-free) would clarify the scope of the Poisson reconstruction.
  4. In (30) the chemical-potential extrapolation uses a convex-splitting-style χ; a short sentence relating this choice to the energy estimate of Theorem 4.6 would make the free-energy telescoping easier to follow.
  5. Typos: “GuWangSGE2025” formatting in the abstract; occasional missing spaces after commas in multi-line displays; “Theorem 2.2” in Remark 2.4 should be Definition 2.2.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: GSGE is an algebraic affine-space construction with independent least-squares uniqueness; self-citations only supply the SGE baseline being generalized and a prior energy-estimate technique.

  1. other [Theorem 4.6 proof (CHNS energy law)]
    "The remaining terms give the stated identity as in [17]."

    The final telescoping of the BDF2 free-energy terms is deferred to the authors’ prior CHNS–BDF2 paper rather than re-derived in full. This is a minor technical self-citation of an independent algebraic estimate, not a load-bearing premise of the GSGE theory; the ZEC cancellation used in the same proof is established in-paper (Prop. 4.5).

full rationale

The central claims are self-contained finite-dimensional linear algebra and geometry. Admissible gauges are defined by the contraction condition (ι_∇S ω)^♯ = L∇E (Def. 2.2); Prop. 2.3 shows they form an affine space by the elementary observation that the condition is affine. Thm. 3.1 proves uniqueness of the A-weighted minimum-Hilbert–Schmidt representative by orthogonal decomposition in the Frobenius product, without importing uniqueness from prior work. Regularization (Thm. 3.2), residual projection (Prop. 3.4), and invariant-preserving gauges (Prop. 3.5) are proved from the same least-squares/Gram construction. The rank-two Jacobi criterion (Prop. 3.7) is standard and applied directly. For NS, discrete free-energy neutrality and the Lie-bracket identity are verified from MAC summation-by-parts and quadratic homogeneity (eqs. 26, Thm. 4.2–4.3), not assumed by citation. For CHNS, λ_A and λ_J are chosen so the residual is ZEC-orthogonal by construction (eqs. 31–32), then mass and energy laws are proved (Prop. 4.5, Thm. 4.6). Self-citations to Gu–Wang SGE [16] and BDF2 [17] supply the baseline scheme being generalized and a technical energy-estimate template (“as in [17]”); neither is a load-bearing uniqueness theorem or fitted input renamed as prediction. No fitted parameters, no self-definitional loop, no smuggled ansatz. Score 1 only for the minor non-load-bearing self-citation in the CHNS energy proof.

Assumptions & free parameters 1 free parameters · 4 assumptions · 3 invented entities

The paper rests on standard finite-dimensional linear algebra and the GENERIC degeneracy conditions, plus the modeling assumption that a compatible spatial discretization preserves free-energy neutrality and the algebraic identities needed for the discrete Jacobi identity. No numerical free parameters are fitted; the three regularization weights ℓ1,ℓ2,ℓ3 are free design choices. The only invented entities are the named gauge constructions themselves, which are purely algebraic.

free parameters (1)
  • regularization weights ℓ1, ℓ2, ℓ3 (and σ = ℓ3 τ²)
    Hand-chosen non-negative parameters that define the differential weight A and the regularized denominator; they control which gauge is selected and whether the formula remains defined at vanishing force. Not fitted to data.
assumptions (4)
  • domain assumption GENERIC degeneracy conditions L∇S = 0 and M∇E = 0 (or their isothermal free-energy counterparts)
    Taken as the definition of a thermodynamically consistent system throughout Sections 1–2; used to guarantee free-energy neutrality of the reversible field.
  • standard math Finite-dimensional real inner-product space setting for the structural theory
    Stated in Section 2.1; continuum PDE formulas are understood only as identities to be preserved by compatible discretizations.
  • domain assumption Compatible MAC / periodic summation-by-parts spatial discretization preserves discrete free-energy neutrality and the algebraic identities used in the Jacobi and energy proofs
    Invoked for Theorems 4.2–4.3 and 4.6; without it the fully discrete GENERIC structure does not follow.
  • standard math Rank-two bivector Y∧Z is Poisson iff Y∧Z∧[Y,Z]=0
    Cited from Poisson-geometry literature (Crainic et al.) and used as Proposition 3.7.
invented entities (3)
  • Generalized skew-gradient embedding (GSGE) / admissible ZEC gauge
    purpose: Name the affine family of two-forms that realize the reversible field by contraction with the thermodynamic force
    Definition 2.2; purely algebraic, no independent physical existence claimed.
  • Operator-weighted minimum-Hilbert–Schmidt gauge ω_A,r
    purpose: Select a unique representative from the affine family by least-squares optimality in a chosen metric A
    Theorem 3.1; algebraic construction.
  • Regularized gauge ω_σ and invariant-preserving gauge ω_A,C
    purpose: Keep the gauge defined at vanishing force and force preservation of prescribed invariants
    Theorem 3.2 and Proposition 3.5; algebraic constructions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized skew-gradient embedding for thermodynamically consistent systems." pith.science (2026). https://pith.science/paper/NBNTPJNO

@misc{pith2026260709617,
  author       = {Pith},
  title        = {Pith review of: Generalized skew-gradient embedding for thermodynamically consistent systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBNTPJNO}},
  note         = {Machine review of arXiv:2607.09617}
}
read the original abstract

The skew-gradient embedding (SGE) framework~\cite{GuWangSGE2025} reformulates a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution in a skew-symmetric operator. In a time-discrete scheme, the profiles defining this operator may be evaluated at previous time levels. The resulting operator remains skew-symmetric, so its contribution to the discrete energy balance vanishes; this explicit treatment often decouples multiphysics systems. We show that this operator is not unique: the admissible gauges form an affine space, and we call the resulting family generalized skew-gradient embeddings (GSGE). For any positive definite metric, least squares selects a unique minimum-Hilbert--Schmidt gauge, and the native metric recovers SGE. This construction also gives regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants. For rank-two gauges, we use a necessary and sufficient Jacobi criterion. Applying this criterion to a compatible MAC discretization of the incompressible Navier--Stokes equations gives a finite-dimensional rank-two Poisson--GENERIC formulation at the semi-discrete level; the implicit midpoint rule preserves this rank-two GENERIC structure at the fully discrete level and satisfies the exact discrete energy law. For the Cahn--Hilliard--Navier--Stokes system, the regularized GSGE--BDF2 scheme preserves mass, dissipates the discrete energy unconditionally, and admits a decoupled implementation.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 linked inside Pith

  1. [1]

    Arnold, R.S

    D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006) 1–155

    2006

  2. [2]

    Bloch, M

    A. Bloch, M. Farré Puiggalí, D. Martín de Diego, Metriplectic Euler–Poincaré equations: smooth and discrete dynamics, Commun. Anal. Mech. 16 (2024) 910–927

    2024

  3. [3]

    Averaged Vector Field

    E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D. O’Neale, B. Owren, G.R.W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the “Averaged Vector Field” method, J. Comput. Phys. 231 (2012) 6770–6789

    2012

  4. [4]

    L. Chen, J. Zhao, A novel second-order linear scheme for the Cahn–Hilliard–Navier– Stokes equations, J. Comput. Phys. 423 (2020) 109782

    2020

  5. [5]

    Crainic, R.L

    M. Crainic, R.L. Fernandes, I. Mărcuţ, Lectures on Poisson Geometry, Graduate Studies in Mathematics, vol. 217, American Mathematical Society, Providence, RI, 2021

    2021

  6. [6]

    Elliott, A.M

    C.M. Elliott, A.M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (1993) 1622–1663

    1993

  7. [7]

    Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, MRS Proc

    D.J. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, MRS Proc. 529 (1998) 39–46

    1998

  8. [8]

    X. Feng, T. Tang, J. Yang, Stabilized Crank–Nicolson/Adams–Bashforth schemes for phase field models, East Asian J. Appl. Math. 3 (2013) 59–80

    2013

Show all 53 references
  1. [9]

    Furihata, T

    D. Furihata, T. Matsuo, Discrete Variational Derivative Method: A Structure- Preserving Numerical Method for Partial Differential Equations, CRC Press, Boca Raton, 2010

    2010

  2. [10]

    Y. Gong, J. Zhao, X. Yang, Q. Wang, Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities, SIAM J. Sci. Comput. 40 (2018) B138–B167

    2018

  3. [11]

    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

  4. [12]

    Y. Gong, J. Zhao, Q. Wang, Arbitrarily high-order linear energy stable schemes for gradient flow models, J. Comput. Phys. 419 (2020) 109610. 15

    2020

  5. [13]

    Y. Gong, Q. Hong, Q. Wang, Supplementary variable method for thermodynamically consistent partial differential equations, Comput. Methods Appl. Mech. Engrg. 381 (2021) 113746

    2021

  6. [14]

    Gonzalez, Time integration and discrete Hamiltonian systems, J

    O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci. 6 (1996) 449–467

    1996

  7. [15]

    Grmela, H.C

    M. Grmela, H.C. Öttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56 (1997) 6620–6632

    1997

  8. [16]

    X.Gu, Q.Wang, Skewgradientembeddingforthermodynamicallyconsistentsystems, arXiv:2509.18601, 2025

    2025

  9. [17]

    X. Gu, Q. Wang, An energy-stable implicit convex-splitting BDF2 scheme for the Cahn–Hilliard–Navier–Stokes equations, arXiv:2026.04204, 2026

    2026

  10. [18]

    X. Gu, G. Ji, Q. Wang, Efficient numerical schemes for a two-phase hydrodynamical model of active liquid crystals and solids, Int. J. Eng. Sci. 227 (2026) 104588

    2026

  11. [19]

    Guillén-González, G

    F. Guillén-González, G. Tierra, On linear schemes for a Cahn–Hilliard diffuse interface model, J. Comput. Phys. 234 (2013) 140–171

    2013

  12. [20]

    Hairer, C

    E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure- Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin, 2006

    2006

  13. [21]

    D. Han, X. Wang, A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn–Hilliard–Navier–Stokes equation, J. Comput. Phys. 290 (2015) 139–156

    2015

  14. [22]

    Harlow, J.E

    F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incom- pressible flow of fluid with free surface, Phys. Fluids 8 (1965) 2182–2189

    1965

  15. [23]

    Q. Hong, Q. Wang, Thermodynamically consistent hybrid computational models for fluid-particle interactions, J. Comput. Phys. 513 (2024) 113147

    2024

  16. [24]

    T. Hou, H. Leng, Numerical analysis of a stabilized Crank–Nicolson/Adams– Bashforth finite difference scheme for Allen–Cahn equations, Appl. Math. Lett. 102 (2020) 106150

    2020

  17. [25]

    Jiang, Z

    M. Jiang, Z. Zhang, J. Zhao, Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation, J. Comput. Phys. 456 (2022) 110954

    2022

  18. [26]

    Jiang, Q

    N. Jiang, Q. Wang, A thermodynamically consistent model for yield stress fluids, arXiv:2406.00813, 2024. 16

    2024 arXiv

  19. [27]

    D. Kay, R. Welford, Efficient numerical solution of Cahn–Hilliard–Navier–Stokes fluids in 2D, SIAM J. Sci. Comput. 29 (2007) 2241–2257

    2007

  20. [28]

    Lipnikov, G

    K. Lipnikov, G. Manzini, M. Shashkov, Mimetic finite difference method, J. Comput. Phys. 257 (2014) 1163–1227

    2014

  21. [29]

    McLachlan, G.R.W

    R.I. McLachlan, G.R.W. Quispel, N. Robidoux, Geometric integration using discrete gradients, Philos. Trans. R. Soc. A 357 (1999) 1021–1045

    1999

  22. [30]

    Morrison, Poisson brackets for fluids and plasmas, AIP Conf

    P.J. Morrison, Poisson brackets for fluids and plasmas, AIP Conf. Proc. 88 (1982) 13–46

    1982

  23. [31]

    Morrison, A paradigm for joined Hamiltonian and dissipative systems, Phys

    P.J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Phys. D 18 (1986) 410–419

    1986

  24. [32]

    Morrison, Hamiltonian description of the ideal fluid, Rev

    P.J. Morrison, Hamiltonian description of the ideal fluid, Rev. Mod. Phys. 70 (1998) 467–521

    1998

  25. [33]

    Onsager, Reciprocal relations in irreversible processes

    L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev. 37 (1931) 405–426

    1931

  26. [34]

    Onsager, Reciprocal relations in irreversible processes

    L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38 (1931) 2265–2279

    1931

  27. [35]

    Öttinger, M

    H.C. Öttinger, M. Grmela, Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E 56 (1997) 6633–6655

    1997

  28. [36]

    Öttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, 2005

    H.C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, 2005

    2005

  29. [37]

    Öttinger, GENERIC integrators: Structure preserving time integration for thermodynamic systems, J

    H.C. Öttinger, GENERIC integrators: Structure preserving time integration for thermodynamic systems, J. Non-Equilib. Thermodyn. 43 (2018) 89–100

    2018

  30. [38]

    J. Shen, X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations, Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691

    2010

  31. [39]

    J. Shen, X. Yang, Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows, SIAM J. Numer. Anal. 53 (2015) 279–296

    2015

  32. [40]

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

    2018

  33. [41]

    Shin, H.G

    J. Shin, H.G. Lee, J.Y. Lee, Unconditionally stable methods for gradient flow using convex splitting Runge–Kutta scheme, J. Comput. Phys. 347 (2017) 367–381

    2017

  34. [42]

    Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001

    R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001

    2001

  35. [43]

    van der Schaft, D

    A. van der Schaft, D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control 1 (2014) 173–378. 17

    2014

  36. [44]

    Wang, Generalized Onsager principle and its applications, in: X.-Y

    Q. Wang, Generalized Onsager principle and its applications, in: X.-Y. Liu (Ed.), Frontiers and Progress of Current Soft Matter Research, Springer, Singapore, 2021, pp. 101–132

    2021

  37. [45]

    J. Yang, J. Kim, On a two-phase incompressible diffuse interface fluid model with curvature-dependent mobility, J. Comput. Phys. 525 (2025) 113764

    2025

  38. [46]

    Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J

    X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys. 327 (2016) 294–316

    2016

  39. [47]

    Yang, A new efficient fully-decoupled and second-order time-accurate scheme for Cahn–Hilliard phase-field model of three-phase incompressible flow, Comput

    X. Yang, A new efficient fully-decoupled and second-order time-accurate scheme for Cahn–Hilliard phase-field model of three-phase incompressible flow, Comput. Methods Appl. Mech. Engrg. 376 (2021) 113589

    2021

  40. [48]

    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. Comput. Phys. 432 (2021) 110015

    2021

  41. [49]

    Z. Yang, S. Dong, An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices, J. Comput. Phys. 393 (2019) 229–257

    2019

  42. [50]

    Zhang, J

    Y. Zhang, J. Shen, A generalized SAV approach with relaxation for dissipative systems, J. Comput. Phys. 464 (2022) 111311

    2022

  43. [51]

    J. Zhao, Q. Wang, X. Yang, Numerical approximations to a new phase field model for two phase flows of complex fluids, Comput. Methods Appl. Mech. Engrg. 310 (2016) 77–97

    2016

  44. [52]

    J. Zhao, X. Yang, Y. Gong, Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg. 318 (2017) 803–825

    2017

  45. [53]

    J. Zhao, D. Han, Second-order decoupled energy-stable schemes for Cahn–Hilliard– Navier–Stokes equations, J. Comput. Phys. 443 (2021) 110536. 18

    2021

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.