Abstract
In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.
| Original language | English |
|---|---|
| Pages (from-to) | 338-356 |
| Number of pages | 19 |
| Journal | Mathematics and Computers in Simulation |
| Volume | 215 |
| DOIs | |
| Publication status | Published - 2024 Jan |
Bibliographical note
Funding Information:The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2C1003844 ). The authors are grateful to the referees whose comments greatly improved the paper.
Publisher Copyright:
© 2023 International Association for Mathematics and Computers in Simulation (IMACS)
Keywords
- Allen–Cahn equation
- Cubic surface
- Diffusion equation
- Finite difference method
ASJC Scopus subject areas
- Theoretical Computer Science
- General Computer Science
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics