Stability analysis for the maximum principle preserving finite difference approach of the surface Allen–Cahn equation on curved surfaces

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Abstract

A stability analysis of a finite difference scheme was presented for the surface Allen–Cahn (AC) equation on curved surfaces. The surface was discretized by a triangular mesh, which accurately represented geometric features and enabled subsequent computational analysis. A discrete Laplace–Beltrami operator was used on this mesh, and it produced a simple and efficient discrete equation. The explicit time-stepping method, although straightforward, requires a moderate restriction on the time step. For practical applications, it is essential to determine the maximum value of this restriction. We derived the theoretical maximum value of the admissible time step that guarantees the discrete maximum principle and analyzed the stability of the explicit scheme under this condition. Numerical tests were performed to verify the theoretical prediction. The computational results confirmed that the proposed scheme maintains stability and reproduces expected interfacial dynamics on curved geometries. The analysis provides rigorous guidance for selecting stable time steps and demonstrates that the fully explicit scheme can be used as a reliable tool for simulating curvature-driven phase transitions and interfacial motion. This work provides a solid theoretical foundation for explicit numerical methods applied to the surface AC equation and contributes to the improvement of both efficiency and accuracy in computational studies involving complex geometries.

Keywords

stability analysis / Laplace–Beltrami operator / maximum principle preserving

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