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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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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DOI: https://doi.org/10.1007/s10444-023-10028-0
Keywords
- Nematic liquid crystal flows
- Virtual element method
- Crank-Nicolson scheme
- Unconditional energy stability
- Error estimates