We develop a linear, first-order, decoupled, energy-stable scheme for a binary hydrodynamic phase field model of mixtures of nematic liquid crystals and viscous fluids that satisfies an energy dissipation law. We show that the semi-discrete scheme in time satisfies an analogous, semi-discrete energy-dissipation law for any time-step and is therefore unconditionally stable. We then discretize the spatial operators in the scheme by a finite-difference method and implement the fully discrete scheme in a simplified version using CUDA on GPUs in 3 dimensions in space and time. Two numerical examples for rupture of nematic liquid crystal filaments immersed in a viscous fluid matrix are given, illustrating the effectiveness of this new scheme in resolving complex interfacial phenomena in free surface flows of nematic liquid crystals.
Zhao, J., Yang, X., Shen, J., & Wang, Q. (2016). A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. Journal of Computational Physics, 305, 539–556. https://doi.org/10.1016/j.jcp.2015.09.044