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Efficient Structure-Preserving Scheme for the Space Fractional Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

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Abstract

In this paper, a numerical method is proposed for solving the space fractional Allen–Cahn equation with logarithmic Flory–Huggins potential. Due to the strong nonlinearity of potential function and singularity of logarithmic term, how to constructed an effective scheme that preserves both the unconditional discrete maximum principle and unconditional original energy stability is a great challenge for this problem. To overcome these difficulties, the stabilized method is applied to construct a structure-preserving scheme, where the weighted and shifted Grünwald difference formula is used to approximate the Riesz fractional derivative. The advantages of this method are that the nonlinear logarithmic term is explicitly treated, the proposed scheme is linear and easy to implement and it preserves the unconditional original energy stability. Then, for any time step, the unique solvability of scheme, discrete maximum principle preserving and original energy stability are all rigorously proved. Moreover, the detailed error estimate in maximum norm is given. In addition, adaptive time-stepping algorithm is used to improve computational efficiency in long time simulations. In numerical experiments, the effects of fractional order \(\alpha \) on the phase change process are also studied. The numerical results verify the effectiveness of the proposed algorithms and the correctness of the theoretical analysis.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China Project (No. 12261131501), the Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department (No. 2023GK2029, No. 2024JC1003, No. 2024JJ1008), and “Algorithmic Research on Mathematical Common Fundamentals” Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China.

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Authors and Affiliations

  1. CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, 518057, China

    Biao Zhang

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, China

    Biao Zhang

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Yin Yang

  4. National Center for Applied Mathematics in Hunan, Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan, 411105, Hunan, China

    Yin Yang

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  1. Biao Zhang
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Zhang, B., Yang, Y. Efficient Structure-Preserving Scheme for the Space Fractional Allen–Cahn Equation with Logarithmic Flory–Huggins Potential. J Sci Comput 103, 18 (2025). https://doi.org/10.1007/s10915-025-02832-1

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  • DOI: https://doi.org/10.1007/s10915-025-02832-1

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