A Strict Physicality-Preserving Scheme for a 2D Q-Tensor Flow with a Singular Potential

Nematic liquid crystals are a state of matter that exhibit properties between those of conventional liquids and solid crystals. Their unique ability to align molecules in specific directions makes them essential in various applications, including display technologies and advanced materials. To model their complex behavior, mathematical frameworks such as the Q-tensor model are used to describe the orientation and degree of molecular order. In this work, we introduce a numerical scheme for a two-dimensional (2D) dynamic Q-tensor model, which is formulated as an L2-gradient flow driven by the liquid crystal free energy and incorporates a singular potential to enforce physical constraints. The proposed scheme is designed to strictly preserve the physicality of the system, ensuring that the fundamental properties of the liquid crystals are respected throughout the simulation. We provide a rigorous convergence analysis of the scheme, establishing the well-posedness of the underlying partial differential equation (PDE) system for the 2D Q-tensor model. To validate its efficiency and accuracy, we perform extensive numerical experiments that demonstrate the scheme’s ability to capture key physical behaviors of nematic liquid crystals. This work highlights the robustness and potential of the proposed scheme for advancing the study and simulation of liquid crystal dynamics.