Efficient Moving Mesh Methods for $Q$-Tensor Models of Nematic Liquid Crystals

Abstract

This paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using $Q$-tensor theory to model the behavior of a nematic liquid crystal cell under the influence of an applied electric field. The key novel feature is the use of a full moving mesh partial differential equation approach to generate an adaptive mesh which accurately resolves important solution features. This includes the use of a new monitor function based on a local measure of biaxiality. In addition, adaptive time-step control is used to ensure the accurate predicting of the switching time, which is often critical in the design of liquid crystal cells. We illustrate the behavior of the method on a one-dimensional time-dependent problem in a Pi-cell geometry which admits two topologically different equilibrium states, modeling the order reconstruction which occurs on the application of an electric field. Our numerical results show that, in addition to achieving optimal rates of convergence in space and time, we obtain higher levels of solution accuracy and a considerable improvement in computational efficiency compared to other moving mesh methods used previously for liquid crystal problems.