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

Hyundong Kim; Jian Wang; Seokjun Ham; Youngjin Hwang; Jyoti; Junseok Kim; Hyundong Kim; Jian Wang; Seokjun Ham; Youngjin Hwang; Jyoti; Junseok Kim

doi:10.3934/era.2025255

Electronic Research Archive

2025, Volume 33, Issue 9: 5748-5768. doi: 10.3934/era.2025255

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

1.
Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
2.
Institute for Smart Infrastructure, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
3.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
4.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
5.
The Institute of Basic Science, Korea University, Seoul 02841, Republic of Korea

Received: 04 August 2025 Revised: 13 September 2025 Accepted: 18 September 2025 Published: 25 September 2025

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.
- stability analysis,
- Laplace–Beltrami operator,
- maximum principle preserving
Citation: Hyundong Kim, Jian Wang, Seokjun Ham, Youngjin Hwang, Jyoti, Junseok Kim. Stability analysis for the maximum principle preserving finite difference approach of the surface Allen–Cahn equation on curved surfaces[J]. Electronic Research Archive, 2025, 33(9): 5748-5768. doi: 10.3934/era.2025255

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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.

References

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