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A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

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Abstract

In this paper, a numerical method is proposed to solve the space fractional Allen-Cahn equation. Based on Crank-Nicolson method for time discretization and second-order weighted and shifted Grünwald difference formula for spatial discretization, we present a new linearized two-level scheme, where the nonlinear term is handled by Newton linearized technology. And we only need to solve a linear system at each time level. Then, the unique solvability of the numerical scheme is given. Under the appropriate assumptions of time step, the discrete maximum principle and energy stability of the numerical scheme are proved. Furthermore, we give a detailed error analysis, which reflects that the temporal and spatial convergence orders are both second order. At last, some numerical experiments show that the proposed method is reasonable and effective.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

Yin Yang is supported by the National Natural Science Foundation of China Project (12071402), the National Key Research and Development Program of China (2020YFA0713503), the Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department (2020JJ2027). Biao Zhang is supported by the Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210594).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan, 411105, Hunan, China

    Biao Zhang

  2. School of Mathematics and Computational Science, Xiangtan University, 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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Correspondence to Yin Yang.

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Zhang, B., Yang, Y. A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation. Numer Algor 93, 179–202 (2023). https://doi.org/10.1007/s11075-022-01411-x

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  • DOI: https://doi.org/10.1007/s11075-022-01411-x

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