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Maximum bound principle for matrix-valued Allen-Cahn equation and integrating factor Runge-Kutta method

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Abstract

The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal matrix-valued functions and has attracted much attention in recent years. It is well known that the MAC equation satisfies the maximum bound principle (MBP) with respect to either the matrix 2-norm or the Frobenius norm, which plays a key role in understanding the physical meaning and the wellposedness of the model. To preserve this property, we extend the explicit integrating factor Runge-Kutta (IFRK) method to the MAC equation. Moreover, we construct a new three-stage third-order and a new four-stage fourth-order IFRK schemes based on the classical Runge-Kutta schemes. Under a reasonable time-step constraint, we prove the MBP preservation of the IFRK method with respect to the matrix 2-norm, based on which, we further establish their optimal error estimates in the matrix 2-norm. Although numerical results indicate that the IFRK method preserves the MBP with respect to the Frobenius norm, a detailed analysis shows that it is hard to prove this preservation by using the same approach for the case of 2-norm. Several numerical experiments are carried out to test the convergence of the IFRK schemes and to verify the MBP preservation with respect to the matrix 2-norm and the Frobenius norm, respectively. Energy stability is also observed, which clearly indicates the orthogonality of the stationary solution of the MAC equation. In addition, we simulate the coarsening dynamics to verify the motion law of the interface for different initial conditions.

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Acknowledgements

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

Funding

This work was partially supported by the National Natural Science Foundation of China (No.12001539), China Postdoctoral Science Foundation (No. 2019TQ0073), and the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20210012).

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  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Yabing Sun & Quan Zhou

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  1. Yabing Sun
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Yabing Sun: Conceptualization, Methodology, Supervision, Writing. Quan Zhou: Methodology, Software.

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Correspondence to Quan Zhou.

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This work was partially supported by the National Natural Science Foundation of China (No. 12001539), China Postdoctoral Science Foundation (No. 2019TQ0073), and the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20210012).

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Sun, Y., Zhou, Q. Maximum bound principle for matrix-valued Allen-Cahn equation and integrating factor Runge-Kutta method. Numer Algor 97, 391–429 (2024). https://doi.org/10.1007/s11075-023-01708-5

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