Close Search Advanced
Journals
Resources
About Us
Open Access

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation
Preview
Add to basket

Year:    2012

Communications in Computational Physics, Vol. 11 (2012), Iss. 4 : pp. 1261–1278

Abstract

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.300810.140411s

Communications in Computational Physics, Vol. 11 (2012), Iss. 4 : pp. 1261–1278

Published online:    2012-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    18

Keywords:   

  1. Error analysis of full-discrete invariant energy quadratization schemes for the Cahn–Hilliard type equation

    Zhang, Jun | Zhao, Jia | Gong, Yuezheng

    Journal of Computational and Applied Mathematics, Vol. 372 (2020), Iss. P.112719

    https://doi.org/10.1016/j.cam.2020.112719 [Citations: 8]

  2. An adaptive time stepping method with efficient error control for second-order evolution problems

    Huang, JianGuo | Lai, JunJiang | Tang, Tao

    Science China Mathematics, Vol. 56 (2013), Iss. 12 P.2753

    https://doi.org/10.1007/s11425-013-4730-x [Citations: 4]

  3. An Adaptive Time Stepping Method for Transient Dynamic Response Analysis

    Huang, Jianguo | Sheng, Huashan

    East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 2 P.152

    https://doi.org/10.4208/eajam.240715.050216a [Citations: 0]

  4. Mass conservative lattice Boltzmann scheme for a three-dimensional diffuse interface model with Peng-Robinson equation of state

    Qiao, Zhonghua | Yang, Xuguang | Zhang, Yuze

    Physical Review E, Vol. 98 (2018), Iss. 2

    https://doi.org/10.1103/PhysRevE.98.023306 [Citations: 8]

  5. Unconditional energy dissipation law and optimal error estimate of fast L1 schemes for a time-fractional Cahn–Hilliard problem

    Huang, Chaobao | An, Na | Yu, Xijun

    Communications in Nonlinear Science and Numerical Simulation, Vol. 124 (2023), Iss. P.107300

    https://doi.org/10.1016/j.cnsns.2023.107300 [Citations: 3]

  6. An Exponential Time Differencing Runge–Kutta Method ETDRK32 for Phase Field Models

    Cao, Weichen | Yang, Hengli | Chen, Wenbin

    Journal of Scientific Computing, Vol. 99 (2024), Iss. 1

    https://doi.org/10.1007/s10915-024-02474-9 [Citations: 0]

  7. Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

    Li, Yang | Sun, Qihang | Feng, Naidan | Liu, Jianjun | Vetro, Calogero

    Journal of Function Spaces, Vol. 2023 (2023), Iss. P.1

    https://doi.org/10.1155/2023/1869660 [Citations: 0]

  8. An Adaptive Time-Stepping Method for the Binary Fluid-Surfactant Phase Field Model on Evolving Surfaces

    Huang, Shijie | Xiao, Xufeng | Feng, Xinlong

    Journal of Scientific Computing, Vol. 95 (2023), Iss. 1

    https://doi.org/10.1007/s10915-023-02150-4 [Citations: 5]

  9. The SCR-Based Adaptive Finite ElementMethod for the Cahn-Hilliard Equation

    田, 文艳

    Advances in Applied Mathematics, Vol. 11 (2022), Iss. 11 P.8355

    https://doi.org/10.12677/AAM.2022.1111884 [Citations: 0]

  10. Asymptotically compatible energy law of the Crank–Nicolson type schemes for time-fractional MBE models

    Zhu, Xiaohan | Liao, Hong-lin

    Applied Mathematics Letters, Vol. 134 (2022), Iss. P.108337

    https://doi.org/10.1016/j.aml.2022.108337 [Citations: 4]

  11. Energy-conserving SAV-Hermite–Galerkin spectral scheme with time adaptive method for coupled nonlinear Klein–Gordon system in multi-dimensional unbounded domains

    Zhang, Xiaohao | Mei, Liquan | Guo, Shimin

    Journal of Computational and Applied Mathematics, Vol. 454 (2025), Iss. P.116204

    https://doi.org/10.1016/j.cam.2024.116204 [Citations: 0]

  12. Energy dissipation law of the variable time-step fractional BDF2 scheme for the time fractional molecular beam epitaxial model

    Zhao, Xuan | Jiang, Zhuhan | Sun, Hong

    International Journal of Computer Mathematics, Vol. 101 (2024), Iss. 9-10 P.1188

    https://doi.org/10.1080/00207160.2024.2315131 [Citations: 1]

  13. A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

    Sun, Jingwei | Zhang, Hong | Qian, Xu | Song, Songhe

    Journal of Computational Physics, Vol. 492 (2023), Iss. P.112414

    https://doi.org/10.1016/j.jcp.2023.112414 [Citations: 5]

  14. Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations

    Duan, Jiayi | Li, Xiao | Qiao, Zhonghua

    Numerical Methods for Partial Differential Equations, Vol. 40 (2024), Iss. 6

    https://doi.org/10.1002/num.23125 [Citations: 0]

  15. Stabilized semi-implicit numerical schemes for the Cahn–Hilliard-like equation with variable interfacial parameter

    Xu, Zhen | Zhang, Hui

    Journal of Computational and Applied Mathematics, Vol. 346 (2019), Iss. P.307

    https://doi.org/10.1016/j.cam.2018.06.031 [Citations: 10]

  16. Efficient and Stable Time Integration of Cahn–Hilliard Equations: Explicit, Implicit, and Explicit Iterative Schemes

    Botchev, M. A. | Fahurdinov, I. A. | Savenkov, E. B.

    Computational Mathematics and Mathematical Physics, Vol. 64 (2024), Iss. 8 P.1726

    https://doi.org/10.1134/S0965542524700945 [Citations: 0]

  17. Asymptotically Compatible Energy and Dissipation Law of the Nonuniform L2-$$1_{\sigma }$$ Scheme for Time Fractional Allen–Cahn Model

    Liao, Hong-lin | Zhu, Xiaohan | Sun, Hong

    Journal of Scientific Computing, Vol. 99 (2024), Iss. 2

    https://doi.org/10.1007/s10915-024-02515-3 [Citations: 3]

  18. High order implicit finite difference schemes with a semi-implicit WENO reconstruction for nonlinear degenerate parabolic equations

    Zhang, Peng | Xiong, Tao

    Journal of Computational Physics, Vol. 467 (2022), Iss. P.111442

    https://doi.org/10.1016/j.jcp.2022.111442 [Citations: 4]

  19. Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations

    Huang, Fukeng | Shen, Jie | Wu, Ke

    Journal of Computational Physics, Vol. 460 (2022), Iss. P.111177

    https://doi.org/10.1016/j.jcp.2022.111177 [Citations: 12]

  20. Energy-stable smoothed particles hydrodynamics for the incompressible single-phase and two-phase flows

    Xiuping, Wang | Tao, Zhang | Shuyu, Sun

    SCIENTIA SINICA Mathematica, Vol. (2024), Iss.

    https://doi.org/10.1360/SSM-2022-0180 [Citations: 0]

  21. Fast algorithm for viscous Cahn-Hilliard equation

    Wang, Danxia | Li, Yaqian | Wang, Xingxing | Jia, Hongen

    Frontiers of Mathematics in China, Vol. 17 (2022), Iss. 4 P.689

    https://doi.org/10.1007/s11464-021-0974-x [Citations: 0]

  22. On a Large Time-Stepping Method for the Swift-Hohenberg Equation

    Zhang, Zhengru | Ma, Yuanzi

    Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 6 P.992

    https://doi.org/10.4208/aamm.2014.m48 [Citations: 14]

  23. Structure-preserving weighted BDF2 methods for anisotropic Cahn–Hilliard model: Uniform/variable-time-steps

    Li, Meng | Bi, Jingjiang | Wang, Nan

    Communications in Nonlinear Science and Numerical Simulation, Vol. 140 (2025), Iss. P.108395

    https://doi.org/10.1016/j.cnsns.2024.108395 [Citations: 0]

  24. Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods

    Krischok, A. | Yaraguntappa, B. | Keip, M.-A.

    Computer Methods in Applied Mechanics and Engineering, Vol. 431 (2024), Iss. P.117220

    https://doi.org/10.1016/j.cma.2024.117220 [Citations: 0]

  25. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

    Feng, Xinlong | Song, Huailing | Tang, Tao | Yang, Jiang

    Inverse Problems & Imaging, Vol. 7 (2013), Iss. 3 P.679

    https://doi.org/10.3934/ipi.2013.7.679 [Citations: 62]

  26. Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

    Yang, Junxiang | Kim, Junseok

    Acta Applicandae Mathematicae, Vol. 172 (2021), Iss. 1

    https://doi.org/10.1007/s10440-021-00405-6 [Citations: 13]

  27. Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation

    Liu, Huan | Cheng, Aijie | Wang, Hong | Zhao, Jia

    Computers & Mathematics with Applications, Vol. 76 (2018), Iss. 8 P.1876

    https://doi.org/10.1016/j.camwa.2018.07.036 [Citations: 85]

  28. Numerical Analysis of a Convex-Splitting BDF2 Method with Variable Time-Steps for the Cahn–Hilliard Model

    Hu, Xiuling | Cheng, Lulu

    Journal of Scientific Computing, Vol. 98 (2024), Iss. 1

    https://doi.org/10.1007/s10915-023-02400-5 [Citations: 0]

  29. An adaptive time-stepping strategy for solving the phase field crystal model

    Zhang, Zhengru | Ma, Yuan | Qiao, Zhonghua

    Journal of Computational Physics, Vol. 249 (2013), Iss. P.204

    https://doi.org/10.1016/j.jcp.2013.04.031 [Citations: 91]

  30. Entropy-Production-Rate-Preserving Algorithms for a Hydrodynamical Model of Binary Fluids

    Sun, Shouwen | Wang, Qi

    Journal of Scientific Computing, Vol. 101 (2024), Iss. 3

    https://doi.org/10.1007/s10915-024-02693-0 [Citations: 0]

  31. On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation

    Zhang, Hong | Zhang, Gengen | Liu, Ziyuan | Qian, Xu | Song, Songhe

    Advances in Computational Mathematics, Vol. 50 (2024), Iss. 3

    https://doi.org/10.1007/s10444-024-10143-6 [Citations: 0]

  32. WO-GRID METHOD FOR BURGERS’ EQUATION BY A NEW MIXED FINITE ELEMENT SCHEME

    Hu, Xiaohui | Huang, Pengzhan | Feng, Xinlong

    Mathematical Modelling and Analysis, Vol. 19 (2014), Iss. 1 P.1

    https://doi.org/10.3846/13926292.2014.892902 [Citations: 7]

  33. Adaptive moving grid methods for two-phase flow in porous media

    Dong, Hao | Qiao, Zhonghua | Sun, Shuyu | Tang, Tao

    Journal of Computational and Applied Mathematics, Vol. 265 (2014), Iss. P.139

    https://doi.org/10.1016/j.cam.2013.09.027 [Citations: 14]

  34. Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction–diffusion equation

    Sun, Hong | Cao, Wanrong | Zhang, Ming

    International Journal of Computer Mathematics, Vol. 100 (2023), Iss. 5 P.991

    https://doi.org/10.1080/00207160.2023.2167517 [Citations: 1]

  35. Mesh-Robustness of an Energy Stable BDF2 Scheme with Variable Steps for the Cahn–Hilliard Model

    Liao, Hong-lin | Ji, Bingquan | Wang, Lin | Zhang, Zhimin

    Journal of Scientific Computing, Vol. 92 (2022), Iss. 2

    https://doi.org/10.1007/s10915-022-01861-4 [Citations: 25]
  36. Geometric Partial Differential Equations - Part I

    The phase field method for geometric moving interfaces and their numerical approximations

    Du, Qiang | Feng, Xiaobing

    2020

    https://doi.org/10.1016/bs.hna.2019.05.001 [Citations: 34]

  37. Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy

    Chen, Wenbin | Li, Weijia | Wang, Cheng | Wang, Shufen | Wang, Xiaoming

    Research in the Mathematical Sciences, Vol. 7 (2020), Iss. 3

    https://doi.org/10.1007/s40687-020-00212-9 [Citations: 29]

  38. Computationally efficient adaptive time step method for the Cahn–Hilliard equation

    Li, Yibao | Choi, Yongho | Kim, Junseok

    Computers & Mathematics with Applications, Vol. 73 (2017), Iss. 8 P.1855

    https://doi.org/10.1016/j.camwa.2017.02.021 [Citations: 40]

  39. Computationally efficient approach for the minimization of volume constrained vector-valued Ginzburg–Landau energy functional

    Tavakoli, Rouhollah

    Journal of Computational Physics, Vol. 295 (2015), Iss. P.355

    https://doi.org/10.1016/j.jcp.2015.04.020 [Citations: 7]

  40. An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

    Guan, Zhen | Heinonen, Vili | Lowengrub, John | Wang, Cheng | Wise, Steven M.

    Journal of Computational Physics, Vol. 321 (2016), Iss. P.1026

    https://doi.org/10.1016/j.jcp.2016.06.007 [Citations: 14]

  41. Fast and stable explicit operator splitting methods for phase-field models

    Cheng, Yuanzhen | Kurganov, Alexander | Qu, Zhuolin | Tang, Tao

    Journal of Computational Physics, Vol. 303 (2015), Iss. P.45

    https://doi.org/10.1016/j.jcp.2015.09.005 [Citations: 33]

  42. An efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation

    Zhao, Shubo | Xiao, Xufeng | Feng, Xinlong

    Applied Mathematics and Computation, Vol. 369 (2020), Iss. P.124901

    https://doi.org/10.1016/j.amc.2019.124901 [Citations: 8]

  43. A benchmark problem for the two- and three-dimensional Cahn–Hilliard equations

    Jeong, Darae | Choi, Yongho | Kim, Junseok

    Communications in Nonlinear Science and Numerical Simulation, Vol. 61 (2018), Iss. P.149

    https://doi.org/10.1016/j.cnsns.2018.02.006 [Citations: 13]

  44. Achieving Convergence in Multiphase Multicomponent Density Gradient Theory Calculations through Regularization

    Maidl, Paul | Langenbach, Kai | Frank, Florian

    Industrial & Engineering Chemistry Research, Vol. 63 (2024), Iss. 32 P.14367

    https://doi.org/10.1021/acs.iecr.4c01669 [Citations: 0]

  45. A non-uniform time-stepping convex splitting scheme for the time-fractional Cahn–Hilliard equation

    Zhang, Jun | Zhao, Jia | Wang, JinRong

    Computers & Mathematics with Applications, Vol. 80 (2020), Iss. 5 P.837

    https://doi.org/10.1016/j.camwa.2020.04.031 [Citations: 16]

  46. Adaptive time-stepping algorithms for molecular beam epitaxy: Based on energy or roughness

    Luo, Fusheng | Xie, Hehu | Xie, Manting | Xu, Fei

    Applied Mathematics Letters, Vol. 99 (2020), Iss. P.105991

    https://doi.org/10.1016/j.aml.2019.07.022 [Citations: 5]

  47. Asymptotically compatible energy of two variable-step fractional BDF2 schemes for the time fractional Allen–Cahn model

    Xing, Zhiyong | Zhang, Haiqing | Liu, Nan

    Applied Mathematics Letters, Vol. 150 (2024), Iss. P.108942

    https://doi.org/10.1016/j.aml.2023.108942 [Citations: 0]

  48. An adaptive finite element method based on Superconvergent Cluster Recovery for the Cahn-Hilliard equation

    Tian, Wenyan | Chen, Yaoyao | Meng, Zhaoxia | Jia, Hongen

    Electronic Research Archive, Vol. 31 (2023), Iss. 3 P.1323

    https://doi.org/10.3934/era.2023068 [Citations: 1]

  49. Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models

    Feng, Xinlong | Tang, Tao | Yang, Jiang

    East Asian Journal on Applied Mathematics, Vol. 3 (2013), Iss. 1 P.59

    https://doi.org/10.4208/eajam.200113.220213a [Citations: 92]

  50. A computational study of lateral phase separation in biological membranes

    Yushutin, Vladimir | Quaini, Annalisa | Majd, Sheereen | Olshanskii, Maxim

    International Journal for Numerical Methods in Biomedical Engineering, Vol. 35 (2019), Iss. 3

    https://doi.org/10.1002/cnm.3181 [Citations: 16]

  51. Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation

    Zhang, Chenhui | Ouyang, Jie | Wang, Xiaodong | Chai, Yong | Ma, Mengxia

    Journal of Scientific Computing, Vol. 87 (2021), Iss. 1

    https://doi.org/10.1007/s10915-021-01430-1 [Citations: 2]

  52. Parameter-Free Time Adaptivity Based on Energy Evolution for the Cahn-Hilliard Equation

    Luo, Fuesheng | Tang, Tao | Xie, Hehu

    Communications in Computational Physics, Vol. 19 (2016), Iss. 5 P.1542

    https://doi.org/10.4208/cicp.scpde14.45s [Citations: 18]

  53. L-stable spectral deferred correction methods and applications to phase field models

    Yao, Lin | Xia, Yinhua | Xu, Yan

    Applied Numerical Mathematics, Vol. 197 (2024), Iss. P.288

    https://doi.org/10.1016/j.apnum.2023.11.020 [Citations: 1]

  54. Operator-splitting methods for the 2D convective Cahn–Hilliard equation

    Gidey, H.H. | Reddy, B.D.

    Computers & Mathematics with Applications, Vol. 77 (2019), Iss. 12 P.3128

    https://doi.org/10.1016/j.camwa.2019.01.023 [Citations: 10]

  55. A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State

    Fan, Xiaolin | Kou, Jisheng | Qiao, Zhonghua | Sun, Shuyu

    SIAM Journal on Scientific Computing, Vol. 39 (2017), Iss. 1 P.B1

    https://doi.org/10.1137/16M1061552 [Citations: 42]

  56. A Highly Efficient and Accurate New Scalar Auxiliary Variable Approach for Gradient Flows

    Huang, Fukeng | Shen, Jie | Yang, Zhiguo

    SIAM Journal on Scientific Computing, Vol. 42 (2020), Iss. 4 P.A2514

    https://doi.org/10.1137/19M1298627 [Citations: 80]

  57. Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method

    Ren, Jin-Lian | Jiang, Rong-Rong | Lu, Wei-Gang | Jiang, Tao

    Acta Physica Sinica, Vol. 69 (2020), Iss. 8 P.080202

    https://doi.org/10.7498/aps.69.20191829 [Citations: 2]

  58. A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation

    Chen, Wenbin | Wang, Xiaoming | Yan, Yue | Zhang, Zhuying

    SIAM Journal on Numerical Analysis, Vol. 57 (2019), Iss. 1 P.495

    https://doi.org/10.1137/18M1206084 [Citations: 82]

  59. Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

    Zhao, Xuan | Xue, Zhongqin

    Communications on Applied Mathematics and Computation, Vol. (2024), Iss.

    https://doi.org/10.1007/s42967-023-00350-1 [Citations: 2]

  60. Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State

    Qiao, Zhonghua | Sun, Shuyu

    SIAM Journal on Scientific Computing, Vol. 36 (2014), Iss. 4 P.B708

    https://doi.org/10.1137/130933745 [Citations: 77]

  61. A Ginzburg-Landau-$${H}^{-1}$$ Model and Its SAV Algorithm for Image Inpainting

    Bai, Xiangyu | Sun, Jiebao | Shen, Jie | Yao, Wenjuan | Guo, Zhichang

    Journal of Scientific Computing, Vol. 96 (2023), Iss. 2

    https://doi.org/10.1007/s10915-023-02252-z [Citations: 6]

  62. Second‐order scalar auxiliary variable Fourier‐spectral method for a liquid thin film coarsening model

    Zhang, Juan | Dong, Lixiu | Zhang, Zhengru

    Mathematical Methods in the Applied Sciences, Vol. 46 (2023), Iss. 18 P.18815

    https://doi.org/10.1002/mma.9594 [Citations: 0]

  63. A three-time-level a posteriori error estimator for GS4-2 framework: Adaptive time stepping for second-order transient systems

    Wang, Yazhou | Xue, Tao | Tamma, Kumar K. | Maxam, Dean | Qin, Guoliang

    Computer Methods in Applied Mechanics and Engineering, Vol. 384 (2021), Iss. P.113920

    https://doi.org/10.1016/j.cma.2021.113920 [Citations: 13]

  64. A linear second-order in time unconditionally energy stable finite element scheme for a Cahn–Hilliard phase-field model for two-phase incompressible flow of variable densities

    Fu, Guosheng | Han, Daozhi

    Computer Methods in Applied Mechanics and Engineering, Vol. 387 (2021), Iss. P.114186

    https://doi.org/10.1016/j.cma.2021.114186 [Citations: 7]

  65. A Novel Lattice Boltzmann Model for Fourth Order Nonlinear Partial Differential Equations

    Qiao, Zhonghua | Yang, Xuguang | Zhang, Yuze

    Journal of Scientific Computing, Vol. 87 (2021), Iss. 2

    https://doi.org/10.1007/s10915-021-01471-6 [Citations: 2]

  66. Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

    Hong, Qi | Gong, Yuezheng | Zhao, Jia | Wang, Qi

    Applied Numerical Mathematics, Vol. 170 (2021), Iss. P.321

    https://doi.org/10.1016/j.apnum.2021.08.002 [Citations: 8]

  67. Unconditionally energy stable time stepping scheme for Cahn–Morral equation: Application to multi-component spinodal decomposition and optimal space tiling

    Tavakoli, Rouhollah

    Journal of Computational Physics, Vol. 304 (2016), Iss. P.441

    https://doi.org/10.1016/j.jcp.2015.10.018 [Citations: 8]

  68. A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

    Liao, Hong-lin | Tang, Tao | Zhou, Tao

    Journal of Computational Physics, Vol. 414 (2020), Iss. P.109473

    https://doi.org/10.1016/j.jcp.2020.109473 [Citations: 101]

  69. Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

    Laadhari, Aymen | Deeb, Ahmad | Kaoui, Badr

    Mathematical Methods in the Applied Sciences, Vol. 46 (2023), Iss. 18 P.19035

    https://doi.org/10.1002/mma.9607 [Citations: 2]

  70. Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

    Guan, Zhen | Lowengrub, John S. | Wang, Cheng | Wise, Steven M.

    Journal of Computational Physics, Vol. 277 (2014), Iss. P.48

    https://doi.org/10.1016/j.jcp.2014.08.001 [Citations: 125]

  71. Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

    Barros, Gabriel F. | Côrtes, Adriano M. A. | Coutinho, Alvaro L. G. A.

    International Journal for Numerical Methods in Engineering, Vol. 122 (2021), Iss. 18 P.5028

    https://doi.org/10.1002/nme.6755 [Citations: 0]

  72. Highly accurate, linear, and unconditionally energy stable large time-stepping schemes for the Functionalized Cahn–Hilliard gradient flow equation

    Zhang, Chenhui | Ouyang, Jie | Wang, Xiaodong | Li, Shuke | Mao, Jiaomin

    Journal of Computational and Applied Mathematics, Vol. 392 (2021), Iss. P.113479

    https://doi.org/10.1016/j.cam.2021.113479 [Citations: 1]

  73. Efficient local energy dissipation preserving algorithms for the Cahn–Hilliard equation

    Mu, Zhenguo | Gong, Yuezheng | Cai, Wenjun | Wang, Yushun

    Journal of Computational Physics, Vol. 374 (2018), Iss. P.654

    https://doi.org/10.1016/j.jcp.2018.08.004 [Citations: 8]

  74. A convex splitting BDF2 method with variable time-steps for the extended Fisher–Kolmogorov equation

    Sun, Qihang | Ji, Bingquan | Zhang, Luming

    Computers & Mathematics with Applications, Vol. 114 (2022), Iss. P.73

    https://doi.org/10.1016/j.camwa.2022.03.017 [Citations: 7]

  75. A Numerical Implementation of the Finite-Difference Algorithm for solving Conserved Cahn–Hilliard Equation

    Boma, Wilcox | Wang, Qinguy | Abiodun, Ayodeji

    Journal of Physics: Conference Series, Vol. 1936 (2021), Iss. 1 P.012014

    https://doi.org/10.1088/1742-6596/1936/1/012014 [Citations: 0]

  76. A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system

    Liu, Zhengguang | Li, Xiaoli

    Journal of Computational Physics, Vol. 447 (2021), Iss. P.110703

    https://doi.org/10.1016/j.jcp.2021.110703 [Citations: 33]

  77. Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation

    Song, Huailing | Shu, Chi-Wang

    Journal of Scientific Computing, Vol. 73 (2017), Iss. 2-3 P.1178

    https://doi.org/10.1007/s10915-017-0497-5 [Citations: 27]

  78. Supplementary variable method for thermodynamically consistent partial differential equations

    Gong, Yuezheng | Hong, Qi | Wang, Qi

    Computer Methods in Applied Mechanics and Engineering, Vol. 381 (2021), Iss. P.113746

    https://doi.org/10.1016/j.cma.2021.113746 [Citations: 29]

  79. An Energy Stable and Maximum Bound Preserving Scheme with Variable Time Steps for Time Fractional Allen--Cahn Equation

    Liao, Hong-lin | Tang, Tao | Zhou, Tao

    SIAM Journal on Scientific Computing, Vol. 43 (2021), Iss. 5 P.A3503

    https://doi.org/10.1137/20M1384105 [Citations: 53]

  80. Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

    Hong, Qi | Zhang, Zengyan | Zhao, Jia

    Journal of Computational Physics, Vol. (2024), Iss. P.113598

    https://doi.org/10.1016/j.jcp.2024.113598 [Citations: 0]

  81. Adaptive time-stepping algorithms for the scalar auxiliary variable scheme of Navier-Stokes equations

    Chen, Hongtao | Wang, Weilong

    Journal of Algorithms & Computational Technology, Vol. 16 (2022), Iss.

    https://doi.org/10.1177/17483026221093956 [Citations: 2]

  82. A Fourier spectral method for fractional-in-space Cahn–Hilliard equation

    Weng, Zhifeng | Zhai, Shuying | Feng, Xinlong

    Applied Mathematical Modelling, Vol. 42 (2017), Iss. P.462

    https://doi.org/10.1016/j.apm.2016.10.035 [Citations: 62]

  83. Multi-phase image segmentation by the Allen–Cahn Chan–Vese model

    Liu, Chaoyu | Qiao, Zhonghua | Zhang, Qian

    Computers & Mathematics with Applications, Vol. 141 (2023), Iss. P.207

    https://doi.org/10.1016/j.camwa.2022.12.020 [Citations: 5]

  84. Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

    Feng, Xinlong | Tang, Tao | Yang, Jiang

    SIAM Journal on Scientific Computing, Vol. 37 (2015), Iss. 1 P.A271

    https://doi.org/10.1137/130928662 [Citations: 73]