Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes

Qiumei Huang; Xiaofeng Yang; Xiaoming He

doi:10.3934/dcdsb.2018230

Article Contents

2018, Volume 23, Issue 6: 2177-2192. Doi: 10.3934/dcdsb.2018230

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Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes

1.
Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China
2.
Department of Mathematics, University of South Carolina, Columbia, SC, 20208, USA
3.
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
4.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

* Corresponding author
* Corresponding author

Received: May 31, 2016

Revised: December 31, 2017

Early access: July 2018

Published: June 2018

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Abstract

In this paper, we consider numerical approximations for a model of smectic-A liquid crystal flows in its weak flow limit. The model, derived from the variational approach of the de Gennes free energy, is consisted of a highly nonlinear system that couples the incompressible Navier-Stokes equations with two nonlinear order parameter equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop an unconditionally energy stable, linear and decoupled time marching numerical scheme for the reduced model in the weak flow limit. We also rigorously prove that the numerical scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability of the scheme.

Keywords:

Mathematics Subject Classification: Primary: 65N35, 65M15, 65M70; Secondary: 65Z05.

Citation:

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Figure 4.1. The $L^2$ errors of the layer funciton $\phi$, the director field ${\bf d} = (d_1, d_2)$, the velocity ${\bf u} = (u, v)$ and pressure $p$. The slopes show that the scheme is asymptotically first-order accurate in time

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Figure 4.2. The evolution of the free energy functional for three different time steps of $\delta t = 0.0001, 0.001$ and $0.01$

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Figure 4.3. Snapshots of the layer function $\phi$ are taken at $t = 0$, $0.2$, $0.4$ and $0.8$ for Example 4.2

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Figure 4.4. Snapshots of the director field ${\bf d}$ are taken at $t = 0$, $0.2$, $0.4$ and $0.8$ for Example 4.2

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Figure 4.5. Time evolution of the free energy functional of Example 4.2

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