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Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods

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Abstract

The energy dissipation law and maximum bound principle are significant characteristics of the Allen-Chan equation. To preserve discrete counterpart of these properties, the linear part of the target system is usually discretized implicitly, resulting in a large linear or nonlinear system of equations. The fast Fourier transform is commonly used to solve the resulting linear or nonlinear systems with computational costs of \(\varvec{\mathcal {O}(M^d \text {log} M)}\) at each time step, where \(\varvec{M}\) is the number of spatial grid points in each direction, and \(\varvec{d}\) is the dimension of the problem. Combining the Saul’yev methods and the stabilization techniques, we propose and analyze novel first- and second-order numerical schemes for the Allen-Cahn equation in this paper. In contrast to the traditional methods, the proposed methods can be solved by components, requiring only \(\varvec{\mathcal {O}(M^d)}\) computational costs per time step. Additionally, they preserve the maximum bound principle and original energy dissipation law at the discrete level. We also propose rigorous analysis of their consistency and convergence. Numerical experiments are conducted to confirm the theoretical analysis and demonstrate the efficiency of the proposed methods.

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Data Availability

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

This work is supported by the National Natural Science Foundation of China (12171245, 11971242).

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Authors and Affiliations

  1. Ministry of Education Key Laboratory for NSLSCS, Jiangsu Collaborative Innovation Center of Biomedical Functional Materials, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Xuelong Gu, Yushun Wang & Wenjun Cai

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  1. Xuelong Gu
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  2. Yushun Wang
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  3. Wenjun Cai
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Correspondence to Wenjun Cai.

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Communicated by: Enrique Zuazua

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Gu, X., Wang, Y. & Cai, W. Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods. Adv Comput Math 50, 34 (2024). https://doi.org/10.1007/s10444-024-10142-7

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