On Accurate and Efficient Numerical Schemes for Energy Based Systems with Applications to Liquid Crystals and Mixtures of Fluids

Abstract

In this work we explore the development, study, and application of numerical algorithms for simulating models represented by partial differential equations for two physical systems: liquid crystals (LCs) and fluid mixtures. The first part focuses on the Landau-de Gennes Q-tensor model to study the properties of nematic, blue phase, and cholesteric LCs. This model is important for understanding LC behavior for a variety of applications in material science, physics, and medicine, where laboratory experiments are often constrained by scale or cost. New numerical schemes are proposed for each system of LCs which preserve the intrinsic structure of the system at the discrete level, while balancing accuracy and computational efficiency through techniques such as linearization and decoupling of the unknowns. The second part investigates phase separation and interfacial dynamics in fluid mixtures of more than two components using a ternary Cahn-Hilliard phase-field model. A new model and new numerical schemes are developed for simulating mixtures of three components, with extensions to larger multi-component systems. This work presents a comprehensive analysis of these two physical systems, the numerical properties of the algorithms, as well as their validation through numerical experiments, alongside the results of interdisciplinary collaborations with material scientists in several applications related to liquid crystals