A stable scheme and its convergence analysis for a 2D dynamic Q-tensor model of nematic liquid crystals

We propose an unconditionally stable numerical scheme for a 2D dynamic Q-tensor model of nematic liquid crystals. This dynamic Q-tensor model is a L2 gradient flow generated by the liquid crystal free energy that contains a cubic term, which is physically relevant but makes the free energy unbounded from below, and for this reason, has been avoided in other numerical studies. The unboundedness of the energy brings significant difficulty in analyzing the model and designing numerical schemes. By using a stabilizing technique, we construct an unconditionally stable scheme, and establish its unique solvability and convergence. Our convergence analysis also leads to, as a byproduct, the well-posedness of the original PDE system for the 2D Q-tensor model. Several numerical examples are presented to validate and demonstrate the effectiveness of the scheme.