In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.
Bibliographical noteFunding Information:
The corresponding author (J. S. Kim) expresses thanks for the support from the BK21 FOUR program. The authors express their sincere gratitude to the reviewers for providing valuable feedback on this revised version. Their constructive comments have significantly contributed to the improvement of the manuscript.
© 2023 the Author(s), licensee AIMS Press
- Allen–Cahn equation
- diffusion equation
- discrete Laplace operator
- triangular meshes
ASJC Scopus subject areas
- General Mathematics