Linear and Decoupled IMEX Schemes for the Ericksen–Leslie Model: Unconditional Stability, Length-Preserving and Rigorous Error Analysis

Liquid crystal hydrodynamics exhibits complex multiscale phenomena governed by strong nonlinearities and anisotropy, posing significant challenges for computational simulation. In this paper, we develop a series of high-order linear and decoupled implicit-explicit (IMEX) schemes for the Ericksen–Leslie model of nematic liquid crystal flows based on the sphere projection. Only two elliptic equations with constant coefficients at each time step need to be solved, which makes the schemes very efficient in simulating such complex fluid systems. A distinctive feature of the proposed IMEX schemes is their ability to assure unconditional energy stability and the preservation of the length of the director field, pivotal for maintaining the physical fidelity of the simulations. We prove that the numerical solutions remain uniformly bounded without constraints on the time step size. Furthermore, we establish rigorous error estimates with orders ranging from 1 to 5 within a unified framework. Several numerical benchmark simulations, including the three-dimensional case that has not been reported in previous studies, are presented to validate the theoretical results of the proposed schemes and to illustrate the defect dynamics of liquid crystal motion.