  In this paper, we develop and investigate numerical methods for the
self-consistent field theory (SCFT) of liquid crystalline polymers. Both the
Flory-Huggins interaction potential and the Maier-Saupe orientational
interaction are considered, enabling simultaneous exploration of microphase
separation and liquid crystalline order in these systems. The main challenge in
numerically solving this complex system lies in solving 6-dimensional (3D
spatial + 2D orientation + 1D time) partial differential equations and
performing nonlinear iterations to optimize the fields. We present ten
numerical algorithms for solving high-dimensional PDEs and give an evaluation
criterion of choosing the most appropriate PDE solver both from theoretical and
numerical performance. Results demonstrate that coupling the fourth-order
Runge-Kutta scheme with Fourier pseudo-spectral methods and spherical harmonic
transformations is superior compared to other algorithms. We develop two
nonlinear iteration schemes, alternating direction iteration method and
Anderson mixing method, and design an adaptive technique to enhance the
stability of the Anderson mixing method. Moreover, the cascadic multi-level
(CML) technique further accelerates the SCFT computation when applied to these
iteration methods. Meanwhile, an approach to optimize the computational domain
is developed. To demonstrate the power of our approaches, we investigate the
self-assembled behaviors of flexible-semiflexible diblock copolymers in
4,5,6-dimensional simulations. Numerical results illustrate the efficiency of
our proposed method.
