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An unconditionally gradient stable numerical method for solving the Allen–Cahn equation

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  • Choi, Jeong-Whan
  • Lee, Hyun Geun
  • Jeong, Darae
  • Kim, Junseok

Abstract

We consider an unconditionally gradient stable scheme for solving the Allen–Cahn equation representing a model for anti-phase domain coarsening in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. We also show the pointwise boundedness of the numerical solution for the Allen–Cahn equation. We describe various numerical experiments we performed to study properties of the Allen–Cahn equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:9:p:1791-1803
    DOI: 10.1016/j.physa.2009.01.026
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Uzunca, Murat & Karasözen, Bülent, 2023. "Linearly implicit methods for Allen-Cahn equation," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    5. 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.
    6. 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.
    7. 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.
    8. 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).

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