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A highly efficient and accurate new SAV approach for the modified phase field crystal model

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Abstract

The modified phase field crystal (MPFC) model is a sixth-order evolutive nonlinear partial differential equation which can describe many crystal phenomena. In this paper, we propose a highly efficient and accurate numerical method to construct linear and unconditionally energy stable schemes for the MPFC model. In recent years, the scalar auxiliary variable (SAV) and SAV-based methods have attracted much attention in numerical solution for dissipative systems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. The considered numerical schemes are based on a new SAV-type approach which named new scalar auxiliary variable (nSAV) approach. We first give a first-order energy stable numerical scheme by introducing a new SAV R(t). Then, the high-order nSAV schemes based on the k-step backward differentiation formula (BDFk) are constructed. The considered nSAV schemes allow us to construct high-order schemes for both the phase variable ϕ and ψ while only using a first-order approximation of the energy balance equation. To our knowledge, there is no careful research to give high-order accurate but energy stable schemes for the MPFC model. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients in one time step which is easy to use fast Fourier transform (FFT) to save more CPU time in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

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Acknowledgements

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

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12001336, 11971276), by the Postdoctoral Science Foundation of China under grant number 2020M672111, and by Shandong Province Natural Science Foundation (Grant No. ZR2020QA030).

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Authors and Affiliations

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

    Zhengguang Liu & Zhaojie Zhou

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

    Nan Zheng

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  1. Zhengguang Liu
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  2. Nan Zheng
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Correspondence to Zhengguang Liu.

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Liu, Z., Zheng, N. & Zhou, Z. A highly efficient and accurate new SAV approach for the modified phase field crystal model. Numer Algor 93, 543–562 (2023). https://doi.org/10.1007/s11075-022-01426-4

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  • DOI: https://doi.org/10.1007/s11075-022-01426-4

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