A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach

Abstract

In this paper, we consider the numerical approximation of a hydrodynamic Ericksen–Leslie system, which describes the macroscopic continuum of liquid crystals. A numerical challenge for solving the governing system is how to construct a suitable numerical scheme so that it is unconditionally energy stable at the discrete level. We propose a novel linear, decoupled fully discrete scheme, which is based on the design of the scalar auxiliary variable (SAV) for the nonlinear potential and combined with the discontinuous Galerkin (DG) for spatial discretization, the pressure-projection method for the Navier–Stokes equations, and implicit–explicit (IMEX) approaches for the highly nonlinear and coupling terms. We rigorously prove the unconditional energy stability and optimal error estimates of the proposed scheme, especially the optimal error bounds for the pressure. Finally, several numerical examples are performed to numerically demonstrate the accuracy, energy stability, and applicability of the proposed scheme.