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Fourier-Spectral Method for the Phase-Field Equations

Author

Listed:
  • Sungha Yoon

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

  • Darae Jeong

    (Department of Mathematics, Kangwon National University, Gangwon-do 24341, Korea)

  • Chaeyoung Lee

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

  • Hyundong Kim

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

  • Sangkwon Kim

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

  • Hyun Geun Lee

    (Department of Mathematics, Kwangwoon University, Seoul 01897, Korea)

  • Junseok Kim

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

Abstract

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1385-:d:400449
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    References listed on IDEAS

    as
    1. Montanelli, Hadrien & Bootland, Niall, 2020. "Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 307-327.
    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. Dongsun Lee & Seunggyu Lee, 2019. "Image Segmentation Based on Modified Fractional Allen–Cahn Equation," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-6, January.
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    Cited by:

    1. 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.

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