IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i1p97-d306090.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/1/97/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/1/97/
    Download Restriction: no
    ---><---

    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. 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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Tan, Zhijun & Yang, Junxiang & Chen, Jianjun & Kim, Junseok, 2023. "An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids," Applied Mathematics and Computation, Elsevier, vol. 438(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Poochinapan, Kanyuta & Wongsaijai, Ben, 2022. "Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    2. Choi, Yongho & Jeong, Darae & Kim, Junseok, 2017. "A multigrid solution for the Cahn–Hilliard equation on nonuniform grids," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 320-333.
    3. Junxiang Yang & Yibao Li & Junseok Kim, 2022. "A Correct Benchmark Problem of a Two-Dimensional Droplet Deformation in Simple Shear Flow," Mathematics, MDPI, vol. 10(21), pages 1-10, November.
    4. Uzunca, Murat & Karasözen, Bülent, 2023. "Linearly implicit methods for Allen-Cahn equation," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    5. Tan, Zhijun & Yang, Junxiang & Chen, Jianjun & Kim, Junseok, 2023. "An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    6. Xiao, Xufeng & Feng, Xinlong, 2022. "A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 36-58.
    7. Qiming Huang & Junxiang Yang, 2022. "Linear and Energy-Stable Method with Enhanced Consistency for the Incompressible Cahn–Hilliard–Navier–Stokes Two-Phase Flow Model," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    8. Sungha Yoon & Darae Jeong & Chaeyoung Lee & Hyundong Kim & Sangkwon Kim & Hyun Geun Lee & Junseok Kim, 2020. "Fourier-Spectral Method for the Phase-Field Equations," Mathematics, MDPI, vol. 8(8), pages 1-36, August.
    9. Koike, Yukito & Nakamula, Atsushi & Nishie, Akihiro & Obuse, Kiori & Sawado, Nobuyuki & Suda, Yamato & Toda, Kouichi, 2022. "Mock-integrability and stable solitary vortices," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    10. Sinhababu, Arijit & Bhattacharya, Anirban, 2022. "A pseudo-spectral based efficient volume penalization scheme for Cahn–Hilliard equation in complex geometries," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 1-24.
    11. Pierluigi Colli & Andrea Signori & Jürgen Sprekels, 2022. "Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 25-58, July.
    12. Ham, Seokjun & Kim, Junseok, 2023. "Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 453-465.
    13. Lee, Hyun Geun & Lee, June-Yub, 2015. "A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 432(C), pages 24-34.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:97-:d:306090. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.