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Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation

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Abstract

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be applied to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. We propose a series of efficient modified stabilized invariant energy quadratization approaches with unconditional energy stability for the phase-field crystal model. Firstly, we propose a more suitable positive preserving function strictly in square root and consider a modified invariant energy quadratization (MIEQ) approach. Secondly, a series of efficient and suitable functionals H(ϕ) in square root are considered and the modified stabilized invariant energy quadratization (MSIEQ) approaches are developed. We prove the unconditional energy stability and optimal error estimates for the semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, MIEQ, and MSIEQ approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

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Acknowledgments

We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.

Funding

The author thanks for the financial support from the China Scholarship Council. This work is supported by the Postdoctoral Science Foundation of China under grant nos. BX20190187 and 2019M650152 and by the National Natural Science Foundation of China (grant nos. 11931003, 41974133, 11901489, 11971276).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, China

    Zhengguang Liu

  2. Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, China

    Xiaoli Li

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  1. Zhengguang Liu
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  2. Xiaoli Li
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Correspondence to Zhengguang Liu.

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Liu, Z., Li, X. Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation. Numer Algor 85, 107–132 (2020). https://doi.org/10.1007/s11075-019-00804-9

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  • DOI: https://doi.org/10.1007/s11075-019-00804-9

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