Stability analysis of an explicit numerical scheme for the Allen-Cahn equation with high-order polynomial potentials

Jaeyong Choi; Seokjun Ham; Soobin Kwak; Youngjin Hwang; Junseok Kim; Jaeyong Choi; Seokjun Ham; Soobin Kwak; Youngjin Hwang; Junseok Kim

doi:10.3934/math.2024941

AIMS Mathematics

2024, Volume 9, Issue 7: 19332-19344. doi: 10.3934/math.2024941

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Stability analysis of an explicit numerical scheme for the Allen-Cahn equation with high-order polynomial potentials

1.
College of Natural & Applied Sciences, University of Guam, 303 University Drive, Mangilao 96923, Guam, USA
2.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Received: 01 April 2024 Revised: 31 May 2024 Accepted: 06 June 2024 Published: 11 June 2024
MSC : 39A14, 65D25, 65N06

The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle.
- Allen-Cahn equation,
- stability analysis,
- finite difference method
Citation: Jaeyong Choi, Seokjun Ham, Soobin Kwak, Youngjin Hwang, Junseok Kim. Stability analysis of an explicit numerical scheme for the Allen-Cahn equation with high-order polynomial potentials[J]. AIMS Mathematics, 2024, 9(7): 19332-19344. doi: 10.3934/math.2024941

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Abstract

The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle.

References

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