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Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation

Author

Listed:
  • Chaeyoung Lee

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Darae Jeong

    (Department of Mathematics, Kangwon National University, Chuncheon-si 200-090, Korea)

  • Junxiang Yang

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Junseok Kim

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

Abstract

We present a nonlinear multigrid implementation for the two-dimensional Cahn–Hilliard (CH) equation and conduct detailed numerical tests to explore the performance of the multigrid method for the CH equation. The CH equation was originally developed by Cahn and Hilliard to model phase separation phenomena. The CH equation has been used to model many interface-related problems, such as the spinodal decomposition of a binary alloy mixture, inpainting of binary images, microphase separation of diblock copolymers, microstructures with elastic inhomogeneity, two-phase binary fluids, in silico tumor growth simulation and structural topology optimization. The CH equation is discretized by using Eyre’s unconditionally gradient stable scheme. The system of discrete equations is solved using an iterative method such as a nonlinear multigrid approach, which is one of the most efficient iterative methods for solving partial differential equations. Characteristic numerical experiments are conducted to demonstrate the efficiency and accuracy of the multigrid method for the CH equation. In the Appendix, we provide C code for implementing the nonlinear multigrid method for the two-dimensional CH equation.

Suggested Citation

  • Chaeyoung Lee & Darae Jeong & Junxiang Yang & Junseok Kim, 2020. "Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation," Mathematics, MDPI, vol. 8(1), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:97-:d:306090
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    References listed on IDEAS

    as
    1. Pierluigi Colli & Gianni Gilardi & Jürgen Sprekels, 2019. "A Distributed Control Problem for a Fractional Tumor Growth Model," Mathematics, MDPI, vol. 7(9), pages 1-32, August.
    2. Junseok Kim & Seunggyu Lee & Yongho Choi & Seok-Min Lee & Darae Jeong, 2016. "Basic Principles and Practical Applications of the Cahn–Hilliard Equation," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-11, October.
    3. Choi, Jeong-Whan & Lee, Hyun Geun & Jeong, Darae & Kim, Junseok, 2009. "An unconditionally gradient stable numerical method for solving the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(9), pages 1791-1803.
    4. Lee, Chaeyoung & Jeong, Darae & Shin, Jaemin & Li, Yibao & Kim, Junseok, 2014. "A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 17-28.
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    Citations

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    Cited by:

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