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arxiv: 2607.00355 · v1 · pith:5D3LKXTOnew · submitted 2026-07-01 · 🧮 math.NA · cs.NA

Relaxed Lagrange Multiplier (RLM) Schemes for Phase Field Models Preserving the Relaxed Original Energy Dissipation Law

Xiaobo Jing , Jia Zhao This is my paper

Pith reviewed 2026-07-02 00:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords phase field modelsenergy dissipation lawLagrange multipliernumerical schemestime discretizationenergy stabilityrelaxed formulation
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The pith

Relaxed Lagrange multiplier schemes dissipate a relaxed version of the original energy in phase field models while remaining uniquely solvable over broad time steps.

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

Phase field models rely on energy functionals that dissipate over time to simulate multiphase phenomena. This paper develops relaxed Lagrange multiplier schemes that dissipate a relaxed form of the original energy and track its dissipation rate, unlike methods that introduce auxiliary variables and modified energies. Adding a relaxation term turns the multiplier into the solution of a scalar quadratic equation that has an explicit closed form. The resulting schemes stay linear, require only two linear solves with constant coefficients per step, and come in first-order and second-order versions that are proven energy stable.

Core claim

The RLM schemes dissipate a relaxed version of the original energy, closely track the original energy dissipation rate, and ensure the resulting discrete system is uniquely solvable over a broad range of time steps. The key step is augmenting the classical Lagrange multiplier formulation with a relaxation term that produces a scalar quadratic equation for the multiplier with an explicit closed-form solution.

What carries the argument

The relaxation term added to the Lagrange multiplier formulation, which produces an explicit closed-form solution for the multiplier via a scalar quadratic equation while preserving energy stability.

If this is right

  • The schemes remain linear and require solving only two linear systems with constant coefficients per time step.
  • Both first-order and second-order variants satisfy the energy stability property.
  • Numerical tests confirm the expected convergence orders and accurate reproduction of interface dynamics.
  • Computational cost per step matches that of scalar auxiliary variable schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same relaxation idea could extend to multiplier-based discretizations of other variational evolution equations.
  • Wider stable time-step ranges could allow longer simulations of multiphase systems without loss of stability.
  • Closer adherence to the original dissipation law may reduce artifacts in applications where auxiliary variables change the underlying dynamics.

Load-bearing premise

Augmenting the Lagrange multiplier formulation with a relaxation term produces a scalar quadratic equation that admits an explicit closed-form solution while preserving energy stability and the ability to track the original dissipation law.

What would settle it

A numerical test on a standard phase field model in which the algebraic system at a time step has no real solution or multiple solutions, or the computed energy fails to follow the claimed relaxed dissipation law.

Figures

Figures reproduced from arXiv: 2607.00355 by Jia Zhao, Xiaobo Jing.

Figure 1
Figure 1. Figure 1: Snapshots of the phase variable ϕ for the Allen–Cahn equation at T = 24, 120, 192, 240. Each row corresponds to a different scheme: SAV-CN (top), RLM-Q-CN with α = 10−2 (middle), and RLM-PC-CN with α = 10−2 (bottom). Each column shows results at the same time point. We conduct quantitative comparisons between these schemes, with the results summarized in [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the energy, scaling factor [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of the relaxed original energy [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Snapshots of the phase variable ϕ for the Cahn–Hilliard equation at T = 500, 5000, 25000, 50000. Each row corresponds to a different scheme: SAV-CN (top), RLM-Q-CN with α = 10−2 (middle), and RLM-PC-CN with α = 10−2 (bottom). Each column shows results at the same time point. To further analyze the behavior of the schemes for the Cahn–Hilliard equation, we focus on the evolution of energy and related quanti… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the energy, scaling factor [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of the original energy E˜ − α(q 2 − 1) and scaling factor q for the Cahn– Hilliard equation with α = 10−2 and 103 . (a) RLM-Q: the energy curves for α = 10−2 coincide with those of α = 103 . (b) RLM-PC: the energy curves for α = 10−2 are also the same as those of α = 103 . (c) and (d) show the time evolutions of q for the two RLM methods with α = 10−2 and α = 103 . Given the high performance… view at source ↗
Figure 7
Figure 7. Figure 7: Long-time simulations of the Cahn–Hilliard equation with the double-well potential using [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Snapshots of the phase variable ϕ for the Cahn–Hilliard equation at T = 1.5 × 105 , 1.5 × 106 , 3×106 . Each row corresponds to a different scheme: RLM-Q-CN with α = 103 , ∆t = 0.1 (top), and RLM-PC-CN with α = 103 , ∆t = 0.1 (bottom). Each column shows results at the same time point. The preceding examples with the double-well potential serve as benchmark cases. We now demonstrate the applicability of RLM… view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the original energies ERLM −α(q 2−1) in (a) and (b), and scaling factor q for the Cahn–Hilliard equation with the Flory–Huggins potential, computed by the RLM-Q-CN and RLM-PC-CN schemes with α = 10−2 and 103 . Next, we compare snapshots of the phase variable at T = 0, 400, 10000, and 20000 obtained by the SAV-CN and RLM-PC-CN schemes. For ∆t = 10−2 , the two methods yield visually indisti… view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots of the phase variable ϕ for polymer solution at T = 0, 400, 10000, 20000. Each row corresponds to a different scheme: SAV-CN (top) and RLM-PC-CN with α = 103 (bottom). Each column shows results at the same time point. Since the constant Csav in the SAV-CN scheme often affects numerical accuracy, we compare two values, Csav = 102 and Csav = 108 , to assess their effects on the energy at different… view at source ↗
Figure 11
Figure 11. Figure 11: Time evolution of the energy (ESAV or ERLM−α(q 2−1)) and the volume drift V (t)−V (0) for the Cahn–Hilliard equation with the Flory–Huggins potential, where V (t) = R Ω ϕ(x, t) dx. (a) Energies of SAV-CN with Csav = 102 (∆t = 10−2 and 5 × 10−2 ), SAV-CN with Csav = 108 (∆t = 5 × 10−2 ), and RLM with α = 103 (∆t = 10−2 and 5 × 10−2 ). (b) Volume drift for the SAV and RLM schemes. 4.4 CH-type Equation with … view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of the energy, scaling factor [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Snapshots of the phase variable ϕ for the liquid thin film model at T = 0, 200, 5000, 10000, computed by the RLM-PC-CN scheme with α = 104 . 4.5 Conserved Allen–Cahn Equation with Double-well Potential in 3D Finally, we consider a three-dimensional binary system with a double-well potential. The governing equation is the conserved Allen–Cahn equation [27] ∂tϕ = −M(µ + L˜), (4.19) µ = −ε 2∆ϕ + ϕ(ϕ 2 − 1), … view at source ↗
Figure 14
Figure 14. Figure 14: Time evolution of the energy, scaling factor [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Snapshots of the phase variable ϕ for the conserved Allen–Cahn equation in 3D at T = 0, 25, 40, 100, computed by the RLM-PC-CN scheme with α = 103 . 5 Conclusion In this paper, we proposed a class of relaxed Lagrange multiplier (RLM) schemes for phase-field models that dissipate a relaxed original energy E˜ = E + α(q 2 − 1) while tracking the original energy dissipation rate when q ≈ 1, and remain linear … view at source ↗
read the original abstract

Phase-field models are typically derived from variational principles for a free-energy functional and are widely used to simulate complex multiphase phenomena in science and engineering. A central goal in designing numerical schemes for these models is to preserve the underlying energy-dissipation law. In this paper, we propose a class of relaxed Lagrange multiplier (RLM) schemes for phase field models. In contrast to popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) methods, which dissipate a modified energy involving auxiliary variables, the RLM schemes dissipate a relaxed version of the original energy and closely track the original energy dissipation rate. Compared with the classical Lagrange multiplier (LM) approach, the RLM schemes ensure that the resulting discrete system is uniquely solvable over a broad range of time steps. The key idea is to augment the LM formulation with a relaxation term, yielding a scalar quadratic equation for the multiplier with an explicit closed-form solution. The resulting schemes are linear and efficient because each time step requires solving only two linear systems with constant coefficients, at a cost comparable to that of SAV schemes. We construct both first-order and second-order variants and prove their energy stability. Numerical experiments verify the expected convergence rates and demonstrate that the RLM schemes accurately capture interface dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes relaxed Lagrange multiplier (RLM) schemes for phase field models. Unlike SAV and IEQ methods that dissipate modified energies, the RLM schemes dissipate a relaxed version of the original energy while closely tracking the original dissipation rate. The key construction augments the classical LM formulation with a relaxation term to produce a scalar quadratic equation for the multiplier that admits an explicit closed-form solution, ensuring the discrete system is uniquely solvable over a broad range of time steps. The resulting schemes are linear, requiring only two linear solves with constant coefficients per step at a cost comparable to SAV. Both first- and second-order variants are constructed and proved energy stable; numerical experiments are reported to confirm convergence rates and accurate capture of interface dynamics.

Significance. If the stated energy-stability proofs and numerical results hold, the contribution would be significant for structure-preserving discretizations of phase-field models. The approach directly addresses two well-known limitations of existing methods: the use of auxiliary energies in SAV/IEQ and the solvability restrictions of classical LM schemes, while retaining linear cost. Explicit closed-form treatment of the multiplier and preservation of a relaxed original energy law would be attractive features for long-time simulations of multiphase phenomena.

major comments (2)
  1. The abstract asserts that energy stability is proved for the first- and second-order RLM schemes and that the schemes 'closely track the original energy dissipation rate,' yet the provided text contains neither the augmented formulation, the quadratic equation, nor any derivation. Without these details the central claims cannot be verified.
  2. The claim that the discrete system is 'uniquely solvable over a broad range of time steps' is load-bearing for the practical advantage over classical LM methods, but no range, no explicit solution formula, and no solvability analysis appear in the available material.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for their careful summary and for highlighting the central claims of the work. We address the major comments point by point below. Only the abstract is available in the provided manuscript material.

read point-by-point responses

  1. Referee: The abstract asserts that energy stability is proved for the first- and second-order RLM schemes and that the schemes 'closely track the original energy dissipation rate,' yet the provided text contains neither the augmented formulation, the quadratic equation, nor any derivation. Without these details the central claims cannot be verified.

    Authors: The abstract is a concise summary and does not contain the technical derivations. The augmented formulation, the quadratic equation for the multiplier, and the energy-stability proofs for the first- and second-order schemes are developed in the body of the full manuscript. Because only the abstract is supplied here, those details cannot be reproduced in this response. revision: no

  2. Referee: The claim that the discrete system is 'uniquely solvable over a broad range of time steps' is load-bearing for the practical advantage over classical LM methods, but no range, no explicit solution formula, and no solvability analysis appear in the available material.

    Authors: The abstract states unique solvability over a broad range of time steps as a distinguishing feature. The explicit closed-form solution, the admissible range of time steps, and the accompanying solvability analysis appear in the full manuscript, which is not included in the provided text. revision: no

standing simulated objections not resolved
  • Augmented formulation, quadratic equation, and energy-stability derivations
  • Explicit solution formula, admissible time-step range, and solvability analysis

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

Only the abstract is available, which describes the RLM method as a direct augmentation of the classical LM formulation by adding a relaxation term to produce a solvable quadratic equation while preserving a relaxed energy dissipation law. No equations, proofs, fitted parameters, or self-citations are supplied that could trigger any of the enumerated circularity patterns. The construction is presented as an independent design choice with separately proven stability, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete information on free parameters, axioms, or invented entities; the relaxation term appears to be an introduced modification but its mathematical status cannot be determined.

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Reference graph

Works this paper leans on

54 extracted references

  1. [1]

    S. M. Allen. Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metallurgica, 20(3):423–433, 1972

    1972

  2. [2]

    D. M. Anderson, G. B. McFadden, and A. A. Wheeler. Diffuse-interface methods in fluid mechanics.Annual Review of Fluid Mechanics, 30(1):139–165, 1998

    1998

  3. [3]

    J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics, 28(2):258–267, 1958

    1958

  4. [4]

    L. Chen, J. Zhao, and X. Yang. Regularized linear schemes for the molecular beam epitaxy model with slope selection.Applied Numerical Mathematics, 128:138–156, 2018

    2018

  5. [5]

    L. Q. Chen and J. Shen. Applications of semi-implicit Fourier-spectral method to phase field equations.Computer Physics Communications, 108:147–158, 1998

    1998

  6. [6]

    Y. Chen, Z. Liu, and X. Meng. Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows.Communications in Nonlinear Science and Numer- ical Simulation, 140:108379, 2025

    2025

  7. [7]

    Cheng, Z

    K. Cheng, Z. Qiao, and C. Wang. A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability.Journal of Scientific Computing, 81(1):154–185, 2019

    2019

  8. [8]

    Cheng, C

    Q. Cheng, C. Liu, and J. Shen. A new Lagrange multiplier approach for gradient flows. Computer Methods in Applied Mechanics and Engineering, 367:113070, 2020

    2020

  9. [9]

    Cheng, J

    Q. Cheng, J. Shen, and C. Wang. Unique solvability and error analysis of a scheme using the Lagrange multiplier approach for gradient flows.SIAM Journal on Numerical Analysis, 63(2):772–799, 2025

    2025

  10. [10]

    Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy.Numerische Mathematik, 63(1):39–65, 1992

    Maria Inˆ es Martins Copetti and Charles M Elliott. Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy.Numerische Mathematik, 63(1):39–65, 1992

    1992

  11. [11]

    C. K. Doan, T. T. P. Hoang, L. Ju, and R. Lan. Dynamically regularized Lagrange multiplier method for the Cahn–Hilliard–Navier–Stokes system.International Journal for Numerical Methods in Engineering, 126(13):e70074, 2025

    2025

  12. [12]

    C. K. Doan, L. Ju, and R. Lan. Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier–Stokes equations.Journal of Computational Physics, 521:113550, 2025

    2025

  13. [13]

    Q. Du, L. Ju, X. Li, and Z. Qiao. Maximum principle preserving exponential time differenc- ing schemes for the nonlocal Allen–Cahn equation.SIAM Journal on Numerical Analysis, 57(2):875–898, 2019

    2019

  14. [14]

    Q. Du, L. Ju, X. Li, and Z. Qiao. Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes.SIAM Review, 63(2):317–359, 2021

    2021

  15. [15]

    H. Egger. Structure preserving approximation of dissipative evolution problems.Numerische Mathematik, 143:85–106, 2019. 34

    2019

  16. [16]

    C. M. Elliott and A. M. Stuart. The global dynamics of discrete semilinear parabolic equations. SIAM Journal on Numerical Analysis, 30(6):1622–1663, 1993

    1993

  17. [17]

    D. J. Eyre. Unconditionally gradient stable time marching the Cahn–Hilliard equation.MRS Online Proceedings Library, 529:39–46, 1998

    1998

  18. [18]

    Furihata and T

    D. Furihata and T. Matsuo.Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, 2011

    2011

  19. [19]

    Y. Gong, Q. Hong, and Q. Wang. Supplementary variable method for thermodynamically consistent partial differential equations.Computer Methods in Applied Mechanics and Engi- neering, 381:113909, 2021

    2021

  20. [20]

    Gong and J

    Y. Gong and J. Zhao. Energy-stable Runge–Kutta schemes for gradient flow models using the energy quadratization approach.Applied Mathematics Letters, 94:224–231, 2019

    2019

  21. [21]

    Guill´ en-Gonz´ alez and G

    F. Guill´ en-Gonz´ alez and G. Tierra. Second order schemes and time-step adaptivity for Allen– Cahn and Cahn–Hilliard models.Computers and Mathematics with Applications, 68(8):821– 846, 2014

    2014

  22. [22]

    D. Hou, Y. Ning, and C. Zhang. An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models.Journal of Computational Physics, 488:112236, 2023

    2023

  23. [23]

    Z. Hu, S. M. Wise, C. Wang, and J. S. Lowengrub. Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation.Journal of Computational Physics, 228(15):5323–5339, 2009

    2009

  24. [24]

    Huang, W

    Q.-A. Huang, W. Jiang, J. Z. Yang, and C. Yuan. A weighted scalar auxiliary variable method for solving gradient flows: Bridging the nonlinear energy-based and Lagrange multiplier ap- proaches.Journal of Scientific Computing, 106(1):16, 2025

    2025

  25. [25]

    Jiang, Z

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

    2022

  26. [26]

    Jiang and J

    M. Jiang and J. Zhao. Linear relaxation schemes for the Allen–Cahn-type and Cahn–Hilliard- type phase field models.Applied Mathematics Letters, 137:108477, 2023

    2023

  27. [27]

    Second order linear energy stable schemes for Allen-Cahn equations with nonlocal constraints.Journal of Scientific Computing, 80(1):500–537, 2019

    Xiaobo Jing, Jun Li, Xueping Zhao, and Qi Wang. Second order linear energy stable schemes for Allen-Cahn equations with nonlocal constraints.Journal of Scientific Computing, 80(1):500–537, 2019

    2019

  28. [28]

    L. Ju, X. Li, Z. Qiao, and H. Zhang. Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection.Mathematics of Computation, 87(312):1859–1885, 2018

    2018

  29. [29]

    Karma and W

    A. Karma and W. Rappel. Quantitative phase-field modeling of dendritic growth in two and three dimensions.Physical Review E, 57:4323–4349, 1998

    1998

  30. [30]

    Li and Z

    D. Li and Z. Qiao. On second order semi-implicit Fourier spectral methods for 2D Cahn– Hilliard equations.Journal of Scientific Computing, 70(1):301–341, 2017

    2017

  31. [31]

    D. Li, Z. Qiao, and T. Tang. Characterizing the stabilization size for semi-implicit Fourier- spectral method to phase field equations.SIAM Journal on Numerical Analysis, 54(3):1653– 1681, 2016. 35

    2016

  32. [32]

    Liu and J

    C. Liu and J. Shen. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method.Physica D: Nonlinear Phenomena, 179(3– 4):211–228, 2003

    2003

  33. [33]

    Liu and X

    Z. Liu and X. Li. A novel Lagrange multiplier approach with relaxation for gradient flows. CSIAM Transactions on Applied Mathematics, 5(1):110–141, 2024

    2024

  34. [34]

    Lowengrub and L

    J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn–Hilliard fluids and topologi- cal transitions.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454(1978):2617–2654, 1998

    1978

  35. [35]

    R. I. McLachlan, G. R. W. Quispel, and N. Robidoux. Geometric integration using discrete gradients.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 357(1754):1021–1045, 1999

    1999

  36. [36]

    Z. Qiao, Z. Zhang, and T. Tang. An adaptive time-stepping strategy for the molecular beam epitaxy models.SIAM Journal on Scientific Computing, 33(3):1395–1414, 2011

    2011

  37. [37]

    J. Shen, C. Wang, X. Wang, and S. M. Wise. Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: Application to thin film epitaxy.SIAM Journal on Numerical Analysis, 50(1):105–125, 2012

    2012

  38. [38]

    J. Shen, J. Xu, and J. Yang. The scalar auxiliary variable (SAV) approach for gradient flows. Journal of Computational Physics, 353:407–416, 2018

    2018

  39. [39]

    J. Shen, J. Xu, and J. Yang. A new class of efficient and robust energy stable schemes for gradient flows.SIAM Review, 61(3):474–506, 2019

    2019

  40. [40]

    Shen and X

    J. Shen and X. Yang. Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete and Continuous Dynamical Systems, 28(4):1669–1691, 2010

    2010

  41. [41]

    Wang and H

    L. Wang and H. Yu. On efficient second order stabilized semi-implicit schemes for the Cahn– Hilliard phase-field equation.Journal of Scientific Computing, 77(2):1185–1209, 2018

    2018

  42. [42]

    S. Wang, W. Wang, and J. Li. An efficient dynamically regularized Lagrange multiplier method for the incompressible MHD equations.Communications in Nonlinear Science and Numerical Simulation, page 109235, 2025

    2025

  43. [43]

    S. L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun, and G. B. McFadden. Thermodynamically-consistent phase-field models for solidification.Physica D: Nonlinear Phenomena, 69:189–200, 1993

    1993

  44. [44]

    S. M. Wise, C. Wang, and J. S. Lowengrub. An energy-stable and convergent finite-difference scheme for the phase field crystal equation.SIAM Journal on Numerical Analysis, 47(3):2269– 2288, 2009

    2009

  45. [45]

    Xu and T

    C. Xu and T. Tang. Stability analysis of large time-stepping methods for epitaxial growth models.SIAM Journal on Numerical Analysis, 44(4):1759–1779, 2006

    2006

  46. [46]

    X. Yang. Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends.Journal of Computational Physics, 327:294–316, 2016. 36

    2016

  47. [47]

    X. Yang, J. Zhao, and Q. Wang. Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method.Journal of Computational Physics, 333:104–127, 2017

    2017

  48. [48]

    X. Yang, J. Zhao, Q. Wang, and J. Shen. Numerical approximations for a three-components Cahn–Hilliard phase-field model based on the invariant energy quadratization method.Math- ematical Models and Methods in Applied Sciences, 27(11):1993–2030, 2017

    1993

  49. [49]

    Zhang, J

    G. Zhang, J. Li, and Q.-A. Huang. A class of unconditionally energy stable relaxation schemes for gradient flows.Mathematics and Computers in Simulation, 218:235–247, 2024

    2024

  50. [50]

    Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model.SIAM Journal on Scientific Computing, 43(2):A1248–A1272, 2021

    Juan Zhang, Cheng Wang, Steven M Wise, and Zhengru Zhang. Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model.SIAM Journal on Scientific Computing, 43(2):A1248–A1272, 2021

    2021

  51. [51]

    Zhang and Z

    Z. Zhang and Z. Qiao. An adaptive time-stepping strategy for the Cahn–Hilliard equation. Communications in Computational Physics, 11(4):1261–1278, 2012

    2012

  52. [52]

    J. Zhao. A revisit of the energy quadratization method with a relaxation technique.Applied Mathematics Letters, 120:107331, 2021

    2021

  53. [53]

    J. Zhao, Q. Wang, and X. Yang. Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach.International Journal for Numerical Methods in Engineering, 110(3):279–300, 2017

    2017

  54. [54]

    J. Zhao, X. Yang, Y. Gong, X. Zhao, J. Li, X. Yang, and Q. Wang. A general strategy for numerical approximations of thermodynamically consistent nonequilibrium models–part I: Thermodynamical systems.International Journal of Numerical Analysis and Modeling, 15(6):884–918, 2018. 37

    2018