A linearized and maximum bound principle preserving finite difference scheme for the Allen–Cahn equation with logarithmic free energy

Baitong Ma; Huiying Hou; Qiuyi Tian; Tianliang Hou; Baitong Ma; Huiying Hou; Qiuyi Tian; Tianliang Hou

doi:10.3934/era.2026064

Electronic Research Archive

2026, Volume 34, Issue 3: 1413-1433. doi: 10.3934/era.2026064

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A linearized and maximum bound principle preserving finite difference scheme for the Allen–Cahn equation with logarithmic free energy

School of Mathematics and Statistics, Beihua University, Jilin 132013, China

Received: 19 December 2025 Revised: 22 January 2026 Accepted: 23 January 2026 Published: 11 February 2026

This study concentrates on the Allen–Cahn equation with logarithmic free energy, thereby proposing a second–order accurate finite difference scheme. The core design of the scheme adopts a linearized modified version of the classical leapfrog scheme for temporal discretization, combined with central differencing for spatial discretization. To further improve the numerical stability, a second–order stabilization term is systematically integrated into the discrete framework. A theoretical analysis reveals that under appropriate constraints on the time step size and stabilization parameter, the numerical solution strictly adheres to the maximum bound principle. A comprehensive stability analysis is performed in the maximum norm, and the corresponding error estimates are rigorously derived. Finally, numerical experiments are conducted to verify the correctness and effectiveness of the theoretical results.
- Allen–Cahn equation,
- leapfrog,
- finite difference method,
- maximum bound principle,
- error estimate
Citation: Baitong Ma, Huiying Hou, Qiuyi Tian, Tianliang Hou. A linearized and maximum bound principle preserving finite difference scheme for the Allen–Cahn equation with logarithmic free energy[J]. Electronic Research Archive, 2026, 34(3): 1413-1433. doi: 10.3934/era.2026064

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Abstract

This study concentrates on the Allen–Cahn equation with logarithmic free energy, thereby proposing a second–order accurate finite difference scheme. The core design of the scheme adopts a linearized modified version of the classical leapfrog scheme for temporal discretization, combined with central differencing for spatial discretization. To further improve the numerical stability, a second–order stabilization term is systematically integrated into the discrete framework. A theoretical analysis reveals that under appropriate constraints on the time step size and stabilization parameter, the numerical solution strictly adheres to the maximum bound principle. A comprehensive stability analysis is performed in the maximum norm, and the corresponding error estimates are rigorously derived. Finally, numerical experiments are conducted to verify the correctness and effectiveness of the theoretical results.

References

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