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

• Published in: Advances in computational mathematics Vol. 49; no. 3
• Main Authors: Zou, Guang-an, Wang, Xuyang, Li, Jian
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2023; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Liquid crystals; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nematic crystals; Stability; Visualization; Nematic liquid crystal flows; Unconditional energy stability; 65M15; Error estimates; Crank-Nicolson scheme; 35Q35; 65M60; 65M12; Virtual element method
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-023-10028-0

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.