向列相液晶流的一种二阶全离散格式
A Second-Order Fully Discrete Scheme for Nematic Liquid Crystal Flow
摘要: 在本文中，我们对向列相液晶流的Ginzburg-Landau模型提出了一种二阶、线性、耦合的格式，证明了该格式在离散条件下的能量稳定性，最后，通过数值模拟展示了四奇异点和旋转流的湮没过程，并且验证了格式的数值精度。结果表明：该格式具有能量稳定性，且具有比较好的数值精度。
Abstract: In this paper, we mainly propose and analyze a second-order, linear, coupled scheme for the Ginzburg-Landau model of the nematic liquid crystal flow, and prove its energy stability under discrete condition. Finally, we demonstrate the annihilation process of four singularities and rotating flows through numerical simulations, and verify the numerical accuracy of the scheme. The results show that the scheme has energy stability and good numerical accuracy.
文章引用：李静, 柴玉珍, 贾宏恩. 向列相液晶流的一种二阶全离散格式[J]. 应用数学进展, 2022, 11(4): 1700-1707. https://doi.org/10.12677/AAM.2022.114185

参考文献

[1] Oseen, C.W. (1933) The Theory of Liquid Crystals. Transactions of the Faraday Society, 29, 883-899. [Google Scholar] [CrossRef] 
[2] Frank, F.C. (1958) Liquid Crystals. On the Theory of Liquid Crystals. Discussions of the Faraday Society, 25, 19-28. [Google Scholar] [CrossRef] 
[3] Adler, J.H., Atherton, T.J., Benson, T.R., Emerson, D.B. and Maclachlan, S.P. (2015) Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects. SIAM Journal on Scientific Computing, 37, S157-S176. [Google Scholar] [CrossRef] 
[4] Adler, J.H., Atherton, T.J., Emerson, D.B. and Maclachlan, S.P. (2015) An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals. SIAM Journal on Numerical Analysis, 53, 2226-2254. [Google Scholar] [CrossRef] 
[5] de Gennes, P.G. (1975) The Physics of Liquid Crystals. Physics Today, 28, 54-55. [Google Scholar] [CrossRef] 
[6] Zhao, J. and Wang, Q. (2016) Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows. Journal of Scientific Computing, 68, 1241-1266. [Google Scholar] [CrossRef] 
[7] Zhao, J., Yang, X., Gong, Y. and Wang, Q. (2017) A Novel Linear Second Order Unconditionally Energy Stable Scheme for a Hydrodynamic Q-Tensor Model of Liquid Crystals. Computer Methods in Applied Mechanics and Engineering, 318, 803-825. [Google Scholar] [CrossRef] 
[8] Cai, Y., Shen, J. and Xu, X. (2017) A Stable Scheme and Its Convergence Analysis for a 2D Dynamic Q-Tensor Model of Nematic Liquid Crystals. Mathematical Models & Methods in Applied Sciences, 27, 1459-1488. [Google Scholar] [CrossRef] 
[9] Ericksen, J.L. (1961) Conservation Laws for Liquid Crystals. Transactions of the Society of Rheology, 5, 23-34. [Google Scholar] [CrossRef] 
[10] Leslie, F.M. (1992) Continuum Theory for Nematic Liquid Crystals. Continuum Mechanics & Thermodynamics, 4, 167-175. [Google Scholar] [CrossRef] 
[11] Ericksen, J.L. (1991) Liquid Crystals with Variable Degree of Orientation. Archive for Rational Mechanics and Analysis, 113, 97-120. [Google Scholar] [CrossRef] 
[12] Badia, S., Guillen-Gonzalez, F. and Gutierrez-Santacreu, J.V. (2011) An Overview on Numerical Analyses of Nematic Liquid Crystal Flows. Archives of Computational Methods in Engineering, 18, Article No. 285. [Google Scholar] [CrossRef] 
[13] Becker, R., Feng, X. and Prohl, A. (2008) Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow. SIAM Journal on Numerical Analysis, 46, 1704-1731. [Google Scholar] [CrossRef] 
[14] Badia, S. and Gutierrez-Santacreu, J.V. (2011) Finite Element Approximation of Nematic Liquid Crystal Flows Using Saddle-Point Structure. Journal of Computational Physics, 230, 1686-1706. [Google Scholar] [CrossRef] 
[15] Labovsky, A., Layton, W.J., Manica, C.C., Neda, M. and Rebholz, L.G. (2009) The Stabilized Extrapolated Trapezoidal Finite-Element Method for the Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering, 198, 958-974. [Google Scholar] [CrossRef] 
[16] Guillen-Gonzalez, F.M. and Gutierrez-Santacreu, J.V. (2013) A Linear Mixed Finite Element Scheme for a Nematic Ericksen-Leslie Liquid Crystal Model. ESAIM: Mathematical Modelling and Numerical Analysis, 47, 1433-1464. [Google Scholar] [CrossRef] 
[17] Cabrales, R.C., Guillen-Gonzalez, F. and Gutierrez-Santacreu, J.V. (2015) A Time-Splitting Finite-Element Stable Approximation for the Ericksen-Leslie Equations. SIAM Journal on Scientific Computing, 37, B261-B282. [Google Scholar] [CrossRef] 
[18] Liu, C. and Walkington, N.J. (2000) Approximation of Liquid Crystal Flows. SIAM Journal on Numerical Analysis, 37, 725-741. [Google Scholar] [CrossRef] 
[19] Ping, L., Liu, C. and Hui, Z. (2007) An Energy Law Preserving C0 Finite Element Scheme for Simulating the Kinematic Effects in Liquid Crystal Dynamics. Journal of Computational Physics, 227, 1411-1427. [Google Scholar] [CrossRef] 

为你推荐