A modified Allen–Cahn equation with a mesh size-dependent interfacial parameter on a triangular mesh

In this article, we propose a modified Allen–Cahn (AC) equation with a space-dependent interfacial parameter. When numerically solving the AC equation with a constant interfacial parameter over large domains, a substantial number of grid points are essential, which leads to significant computational costs. To effectively resolve this problem, numerous adaptive mesh techniques have been developed and implemented. These methods use locally refined meshes that adaptively track the interfacial positions of the phase field throughout the simulation. However, the data structures for adaptive algorithms are generally complex, and the problems to be solved may involve challenges at multiple scales. In this article, we present a modified AC equation with a mesh size-dependent interfacial parameter on a triangular mesh to efficiently solve multi-scale problems. In the proposed method, a triangular mesh is used, and the interfacial parameter value at a node point is defined as a function of the average length of the edges connected to the node point. The proposed algorithm effectively uses large and small values of the interfacial parameter on coarse and fine meshes, respectively. To demonstrate the efficiency and superior performance of the proposed method, we conduct several representative numerical experiments. The computational results indicate that the proposed interfacial function can adequately evolve the multi-scale phase interfaces without excessive relaxation or freezing of the interfaces. Finally, we provide the main source code for the methodology, including mesh generation as described in this paper, so that interested readers can use it.