Consistency-enhanced modified SAV time-stepping method with relaxation for binary mixture of and viscous fluids

In this work, the binary mixture of nematic liquid crystals and viscous fluids (NLC-VF) system consists of the Cahn–Hilliard (CH) equations for the phase-field variable for the free interface, the Allen–Cahn (AC) type constitutive equation for the nematic director, and the incompressible Navier–Stokes (NS) equation for the two fluids. To address the computational challenges posed by this complex system, we propose a scheme that is fully decoupled, energy-stable and highly consistent, based on a modified scalar auxiliary variable (MSAV) with stabilization. Our approach employs two auxiliary variables: one derived from the nonlinear terms in the original energy, and the other leveraging the “zero-energy-contribution (ZEC) ”property satisfied by some nonlinear terms. To ensure consistency between the continuous and discrete auxiliary variables, we employ relaxation techniques. Besides, the proposed method enables sequential solving of each variable, and the computational overhead of the relaxation technique is minimal, resulting in highly efficient computation. We demonstrate the accuracy, stability, consistency, and practicality of the method through comprehensive numerical experiments, and provide rigorous proof of its unconditional energy stability.