Stabilized leapfrog scheme preserving the maximum bound principle for the generalized Allen–Cahn equation

Chunjuan Hou; Baitong Ma; Chunjuan Hou; Baitong Ma

doi:10.3934/era.2025335

Electronic Research Archive

2025, Volume 33, Issue 12: 7584-7599. doi: 10.3934/era.2025335

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Stabilized leapfrog scheme preserving the maximum bound principle for the generalized Allen–Cahn equation

Chunjuan Hou ¹,
Baitong Ma ^{2
,
,}

1.
Institute of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China
2.
School of Mathematics and Statistics, Beihua University, Jilin 132013, China

Received: 10 November 2025 Revised: 08 December 2025 Accepted: 10 December 2025 Published: 18 December 2025

This paper addresses the numerical method for the generalized Allen–Cahn equation featuring nonlinear mobility and a convection term. We propose a linear second–order finite difference scheme that adheres to the discrete maximum bound principle (MBP). The scheme is discretized using the leapfrog finite difference approach, incorporating a stabilized term in time, an upwind scheme for the convection term, and a central–difference scheme for the diffusion term. It is demonstrated that the discrete MBP holds under reasonable constraints on both the time step size and the coefficient of the stabilized term. Additionally, we provide an $ L^\infty\text{–error} $ estimate for our proposed scheme. Several numerical experiments are conducted to validate our theoretical findings.
- generalized Allen–Cahn equation,
- finite difference method,
- maximum bound principle,
- error estimates
Citation: Chunjuan Hou, Baitong Ma. Stabilized leapfrog scheme preserving the maximum bound principle for the generalized Allen–Cahn equation[J]. Electronic Research Archive, 2025, 33(12): 7584-7599. doi: 10.3934/era.2025335

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Abstract

This paper addresses the numerical method for the generalized Allen–Cahn equation featuring nonlinear mobility and a convection term. We propose a linear second–order finite difference scheme that adheres to the discrete maximum bound principle (MBP). The scheme is discretized using the leapfrog finite difference approach, incorporating a stabilized term in time, an upwind scheme for the convection term, and a central–difference scheme for the diffusion term. It is demonstrated that the discrete MBP holds under reasonable constraints on both the time step size and the coefficient of the stabilized term. Additionally, we provide an $ L^\infty\text{–error} $ estimate for our proposed scheme. Several numerical experiments are conducted to validate our theoretical findings.

References

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