A geometric reformulation and structure-preserving rotational discrete gradient scheme for the full Ericksen-Leslie model

Abstract: In this paper, we present a structure-preserving rotational discrete gradient method of the full Ericksen-Leslie model for nematic liquid crystal flows with general anisotropic Oseen-Frank elasticity and full Leslie stress. The main difficulty resides in preserving both the pointwise unit-length constraint on the director field and the energy-dissipation at the discrete level, which involves high nonlinearities in the interplay between the length constraint, anisotropic elastic energy, and hydrodynamic couplings. To address this issue, we first reformulate the Ericksen-Leslie system into an equivalent rotational form in which the director evolution is intrinsically tangent to the unit sphere and the coupling terms in the Leslie and Ericksen stresses are reorganized so that the energy exchange and dissipation structure become explicit. Based on this reformulation, we construct rotational discrete gradient schemes that preserve the director length and satisfy an unconditional discrete energy law, together with a fully discrete scheme based on an exact divergence-free spectral approximation. Numerical experiments verify the accuracy, unit-length preservation and energy-stability of the proposed method, and illustrate dynamical effects induced by anisotropic elastic coefficients and shear flow in the full Ericksen-Leslie model.