Length preserving numerical schemes for the nematic liquid crystal flows

In this paper, we introduce two novel classes of length preserving schemes for the Ericksen– Leslie model based on two distinct Lagrange multiplier approaches. In the first approach, the Lagrange multiplier q ( x , t ) is defined as |∇ d | 2 at the continuous level. In the second approach, q ( x , t ) is introduced at the discrete level to enforce the length constraint, while it remains identically zero at the continuous level. Within a predictor-corrector framework, we develop efficient and robust second-order length preserving schemes for the Ericksen–Leslie model. A series of numerical experiments are presented to verify the accuracy, stability, and energy dissipation properties of the proposed schemes. In addition, numerical comparisons show that our schemes achieve significantly improved computational efficiency compared to the existing method.