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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu

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Volume 20, Issue 2


Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation

Seunggyu Lee
Published Online: 2019-01-26 | DOI: https://doi.org/10.1515/ijnsns-2017-0278


We propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.

Keywords: Cahn–Hilliard equation; finite difference; compact scheme; energy stability; solvability; mass conservation

MSC 2010: 35K35; 65M06; 65M12


  • [1]

    J. Cahn, J. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28 (2) (1958), 258–267.Google Scholar

  • [2]

    D. Lee, J.-Y. Huh, D. Jeong, J. Shin, A. Yun, J. Kim, Physical, mathematical, and numerical derivations of the Cahn–Hilliard equations, Comput. Mater. Sci. 81 (2014), 216–255.

  • [3]

    A. Bertozzi, S. Esedoglu, A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE T. Image Process. 16 (1) (2007), 285–291.

  • [4]

    M. Burger, L. He, C. Schönlieb, Cahn–Hilliard inpainting and a generalization of grayvalue images, SIAM J. Imaging Sci. 2 (4) (2009), 1129–1167.

  • [5]

    E. Khain, L. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E. 77 (2008), 051129.

  • [6]

    M. Burger, R. Stainko, Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim. 45 (4) (2006), 1447–1466.

  • [7]

    S. Zhou, M. Wang, Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition, Struct. Multidiscip. O. 33 (2) (2007), 89–111.

  • [8]

    D. Jeong, S. Lee, Y. Choi, J. Kim, Energy-minimizing wavelengths of equilibrium states for diblock copolymers in the hex-cylinder phase, Curr. Appl. Phys. 15 (2015), 799–804.

  • [9]

    D. Jeong, J. Shin, Y. Li, Y. Choi, J.-H. Jung, S. Lee, J. Kim, Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers, Curr. Appl. Phys. 14 (2014), 1263–1272.

  • [10]

    S. Hu, L. Chen, A phase-field model for evolving microstructures with strong inhomogeneity, Acta Mater. 49 (11) (2001), 1879–1890.

  • [11]

    D. Jeong, S. Lee, J. Kim, An efficient numerical method for evolving microstructures with strong elastic inhomogeneity, Model. Simul. Mater. Sci. Eng. 23 (2015), 045007.

  • [12]

    S. Lee, Y. Choi, D. Lee, H.-K. Jo, S. Lee, S, Myung, J. Kim, A modified Cahn–Hilliard equation for 3d volume reconstruction from two planar cross sections, J. KSIAM. 19 (1) (2015), 47–56.

  • [13]

    Y. Li, J. Shin, Y. Choi, J. Kim, Three-dimensional volume reconstruction from slice data using phase-field models, Comput. Vis. Image Und. 137 (2015), 115–124.Google Scholar

  • [14]

    V. Cristini, X. Li, J. Lowengrub, S. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching, J. Math. Biol. 58 (2009), 723–763.

  • [15]

    X.Wu, G. Zwieten, K. Zee, Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models, Int. J. Numer. M. Biomed. Eng. 30 (2) (2014), 180–203.

  • [16]

    C. Elliott, D. French, Numerical studies of the Cahn–Hilliard equation for phase separation, IMA J. Appl. Math. 38 (2) (1987), 97–128.

  • [17]

    D. Eyer, Unconditionally gradient stable scheme marching the Cahn–Hilliard equation, MRS Proceedings 529 (1998), 39–46.

  • [18]

    Y. He, Y. Liu, T. Tang, On large time-stepping methods for the Cahn–Hilliard equation, Appl. Numer. Math. 57 (5) (2007), 616–628.

  • [19]

    C. Lee, D. Jeong, J. Shin, Y. Li, J. Kim, A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation, Physica A 409 (2014), 17–28.

  • [20]

    S. Lee, C. Lee, H. Lee, J. Kim, Comparison of different numerical schemes for the Cahn–Hilliard equation, J. KSIAM. 17 (3) (2013), 197–207.

  • [21]

    Y. Li, H. Lee, B. Xia, J. Kim, A compact forth-order finite difference scheme for the three-dimensional Cahn–Hilliard eqatuion, Comput. Phys. Commun. 200 (2016), 108–116.

  • [22]

    E. de Mello, O. da Silveira Filho, Numerical study of the Cahn–Hilliard equation in one, two, and three dimensions, Physica A 347 (2005), 429–443.

  • [23]

    T. Rogers, R. Deasi, Numerical study of late-stage coarsening for off-critical quenches in the Cahn–Hilliard equation of phase separation, Phys. Rev. B. 39 (16) (1989), 11956.

  • [24]

    J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, J. Comput. Phys. 55 (1) (1984), 65–80.

  • [25]

    Z. Tian, S. Dai, High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2) (2007), 952–974.

  • [26]

    J. Zhang, Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization, J. Comput. Phys. 179 (1) (2002), 170–179.

  • [27]

    M. Li, T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comput. 16 (1) (2001), 29–45.

  • [28]

    E. Turkel, D. Gordon, R. Gorgon, S. Tsynkov, Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys. 232 (1) (2013), 272–287.

  • [29]

    J. Li, Z. Sun, X. Zhao, A three level linearized compact difference scheme for the Cahn–Hilliard equation, Sci. China Math. 55 (4) (2012), 805–826.

  • [30]

    Q. Du, R. Nicolaides, Numerical analysis of a continuum model of a phase transition. SIAM Numer. Anal. 28 (1991), 1310–1322.

  • [31]

    K. Atkinson, An introduction to numerical analysis, 2nd Edition, John Wiley & Sons, New York, 1988.Google Scholar

  • [32]

    L. Cherfils, A. Miranville, S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561–596.

  • [33]

    J. Greer, A. Bertozzi, G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys. 216 (2006), 216–246.

  • [34]

    U. Trottenberg, C. Oosterlee, A. Schüller, Multigrid, Academic Press, London, 2001.

About the article

Received: 2017-12-18

Accepted: 2019-01-12

Published Online: 2019-01-26

Published in Print: 2019-04-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 137–143, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0278.

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