<p>We propose a numerical algorithm for efficiently solving the Allen–Cahn (AC) equation on large computational domains using a non-uniform mesh. The method enhances spatial resolution near interfaces and reduces degrees of freedom in less active regions so that it achieves a balance between accuracy and efficiency. This adaptive mesh strategy enables refined interface tracking without incurring the high cost of globally fine meshes. A fully explicit Euler scheme with a finite difference method (FDM) is used for time discretization, and it provides simplicity and ease of implementation. For spatial discretization on non-uniform grids, special treatment is given to hanging nodes, where grid points do not align with the structured mesh. In such cases, interpolation from neighboring nodes is applied to maintain stability and consistency. The framework can be extended by introducing variable mobility into the AC formulation so that spatially dependent mobility can better capture physical properties and further improve efficiency. Numerical experiments confirm the method’s stability and accuracy and show that non-uniform grids significantly enhance computational performance while preserving solution fidelity. These results highlight the suitability of the proposed approach for large-scale simulations of phase-field models, particularly in applications dominated by complex interface dynamics.</p>