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An extrapolated Crank-Nicolson virtual element scheme for the nematic liquid crystal flows

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Abstract

In this paper, we consider the numerical approximations of the Ericksen-Leslie system for nematic liquid crystal flows, which can be used to describe the dynamics of low molar-mass nematic liquid crystal in certain materials. The main numerical challenge to solve this system lies in how to discretize nonlinear terms so that the energy stability can be held at the discrete level. This paper address this numerical problem by constructing a fully discrete virtual element scheme with second-order temporal accuracy, which is achieved by combining the extrapolated Crank-Nicolson (C-N) time-stepping scheme for the nonlinear coupling terms and the convex splitting method for the Ginzburg-Landau term. The unconditional energy stability and unique solvability of the fully discrete scheme are rigorously proved, we further prove the optimal error estimates of the developed scheme. Finally, some numerical experiments are presented to demonstrate the accuracy, energy stability, and performance of the proposed numerical scheme.

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Acknowledgements

The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

Funding

Guang-an Zou is supported by China Postdoctoral Science Foundation (No. 2019M662476), and the Key Scientific Research Projects of Colleges and Universities in Henan Province, China (23A110006). Jian Li is supported by NSF of China (No. 11771259), Shaanxi Provincial Joint Laboratory of Artificial Intelligence (No. 2022JC-SYS-05), Innovative team project of Shaanxi Provincial Department of Education (No. 21JP013) and 2022 Shaanxi Provincial Social Science Fund Annual Project (No. 2022D332).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Guang-an Zou & Xuyang Wang

  2. Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, People’s Republic of China

    Guang-an Zou

  3. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Jian Li

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  1. Guang-an Zou
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Communicated by: Long Chen

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Zou, Ga., Wang, X. & Li, J. An extrapolated Crank-Nicolson virtual element scheme for the nematic liquid crystal flows. Adv Comput Math 49, 30 (2023). https://doi.org/10.1007/s10444-023-10028-0

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