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

被引:3
|
作者
Choi, Jaeyong [1 ]
Ham, Seokjun [2 ]
Kwak, Soobin [2 ]
Hwang, Youngjin [2 ]
Kim, Junseok [2 ]
机构
[1] Univ Guam, Coll Nat & Appl Sci, 303 Univ Dr, Mangilao, GU 96923 USA
[2] Korea Univ, Dept Math, Seoul 02841, South Korea
来源
AIMS MATHEMATICS | 2024年 / 9卷 / 07期
关键词
Allen-Cahn equation; stability analysis; finite difference method; RUNGE-KUTTA SCHEMES;
D O I
10.3934/math.2024941
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页码:19332 / 19344
页数:13
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