A second-order accurate numerical scheme for Ericksen-Leslie system with penalty terms and defect dynamics in flows of nematic liquid crystals

In this article, we study a second-order numerical scheme for the Ericksen-Leslie system. A modified Crank-Nicolson temporal discretization is adopted, and the pressure projection approach is used to decouple the Stokes solver. The second-order convex-concave decomposition is applied to the chemical potential vector, in which the convex nonlinear term is updated by a modified Crank-Nicolson approximation, the expansive concave term is computed by an explicit Adams-Bashforth extrapolation, and the surface diffusion part is discretized by the standard Crank-Nicolson interpolation. Meanwhile, semi-implicit second-order approximations are taken for the convection terms, in both the momentum equation and the orientation vector evolutionary equation, as well as the coupled elastic stress terms. In fact, these semi-implicit terms could be represented as a monotone, linear operator of a vector potential, and its combination with the second order convex splitting approximation to the chemical potential yields a closed nonlinear system of equations for the orientation vector function. The unique solvability analysis of the numerical system is established with the help of the Browder-Minty lemma. A combination of the second order convex splitting approximation to chemical potential vector and the semi-implicit discretization for the coupled terms leads to the total energy stability estimate, composed of kinematic energy and internal elastic free energy. Moreover, an optimal rate convergence analysis and error estimate are provided, with second-order accuracy in both time and space. Some numerical simulation results of liquid crystals' molecular dynamics are presented to study the scientific issue of defect annihilation. These results are consistent with the existing literature, and the energy evolution in the defect annihilation process is in agreement with the expectations. In addition, the influence of different parameters on the defect annihilation time instant is tested, and the results also agree with the physical laws.