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

An extrapolated Crank-Nicolson virtual element scheme for the nematic liquid crystal flows 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Computational Mathematics Springer Journals

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

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References (55)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
1019-7168
eISSN
1572-9044
DOI
10.1007/s10444-023-10028-0
Publisher site
See Article on Publisher Site

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.

Journal

Advances in Computational MathematicsSpringer Journals

Published: Jun 1, 2023

Keywords: Nematic liquid crystal flows; Virtual element method; Crank-Nicolson scheme; Unconditional energy stability; Error estimates; 35Q35; 65M12; 65M15; 65M60

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