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Energy Stable Numerical Schemes for a Hydrodynamic Model of Nematic Liquid Crystals

We develop a first-order and a second-order, coupled, energy stable numerical schemes for a modified Ericksen--Leslie hydrodynamic model for nematic liquid crystals. We then discuss two ways to develop decoupled schemes for the model and show that they are energy stable as well. The coupled schemes are implemented in 2-dimensional space, with which we study defect dynamics in flows of nematic liquid crystals. Comparisons of our model predictions with that of a reduced model previously studied, which used the material derivative in place of the time invariant derivative in the Ericksen--Leslie model, are made, demonstrating quite different, but more realistic orientational dynamics in flows of nematic liquid crystals.