Error Estimate for a Decoupled Energy-Stable Scheme for Smectic-a Liquid Crystals

We study a numerical scheme for a hydrodynamic model of smectic-A liquid crystals, which incorporates velocity–pressure variables (u,p) and the order parameter of smectic-A. The model is a highly nonlinear system that couples the incompressible Navier–Stokes equations with a fourth-order parabolic equation for the layer variable ϕ , which is endowed with periodic boundary conditions. We first introduce a first-order decoupled scalar auxiliary variable (SAV) scheme for numerically solving the smectic-A liquid crystal model and present an efficient implementation. Moreover, we prove that the scheme is uniquely solvable and energy stable. Furthermore, we carry out an error estimate for the scheme. Various numerical experiments are presented to verify our theoretical results and simulate the dynamics of the system.