High-accurate and efficient numerical algorithms for the self-consistent field theory of liquid-crystalline polymers

Self-consistent field theory (SCFT) is one of the most widely-used frameworks in studying the equilibrium phase behavior of inhomogeneous polymers. For liquid-crystalline polymeric systems, the primary numerical challenges in solving SCFT involve efficiently solving a large number of 6-dimensional (6D, 3D space + 2D orientation + 1D contour) partial differential equations (PDEs), accurately determining subtle energy differences between self-assembled structures, and developing effective iterative methods for nonlinear SCFT iterations. To address these challenges, this work introduces a suite of high-order and efficient numerical methods tailored to SCFT of liquid-crystalline polymers. These methods include various advanced PDE solvers, an improved Anderson iteration algorithm to accelerate SCFT calculations, and an optimization technique for adjusting the computational domain during SCFT iterations. Extensive numerical tests demonstrate the efficiency of the proposed methods. Based on these algorithms, we further explore the self-assembly behavior of liquid-crystalline polymers through 4D, 5D, and 6D simulations, uncovering intricate 3D spatial structures.