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Up to eighth-order maximum-principle-preserving methods for the Allen–Cahn equation

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Abstract

In this work, we develop a class of up to eighth-order maximum-principle-preserving (MPP) methods for the Allen–Cahn equation. Beginning with the space-discrete system, we extend the integrating factor two-step Runge–Kutta (IFTSRK) methods and define sufficient conditions for the preservation of the discrete maximum principle. In particular, we combine the IFTSRK methods with the linear stabilization technique to develop the stabilized IFTSRK formulations and successfully derive sufficient conditions to preserve the discrete maximum principle unconditionally. Furthermore, we provide error estimates for these proposed methods. Numerical experiments are carried out to illustrate the high-order accuracy and MPP characteristic of the proposed methods and to verify the efficiency through simulations of the long-time evolutional behavior.

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Acknowledgements

The authors sincerely thank the anonymous reviewers for their valuable suggestions, which helped improve this manuscript.

Funding

This research is supported by the National Key R&D Program of China (2020YFA0709803), the National Natural Science Foundation of China (No. 11901577, 11971481, 12071481), the Natural Science Foundation of Hunan (No. 2020JJ5652), the National Key Project (No.GJXM92579), the Defense Science Foundation of China (2021-JCJQ-JJ-0538) and the Research Fund of National University of Defense Technology (No. ZK19-37).

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  1. Department of Mathematics, College of Liberal Arts and Science, National University of Defense Technology, Changsha, Hunan, 410073, China

    Jingwei Sun, Hong Zhang, Xu Qian & Songhe Song

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  1. Jingwei Sun
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  2. Hong Zhang
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Correspondence to Hong Zhang.

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Sun, J., Zhang, H., Qian, X. et al. Up to eighth-order maximum-principle-preserving methods for the Allen–Cahn equation. Numer Algor 92, 1041–1062 (2023). https://doi.org/10.1007/s11075-022-01329-4

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