A UNIFIED FRAMEWORK OF THE SAV-ZEC METHOD FOR A MASS-CONSERVED ALLEN--CAHN TYPE TWO-PHASE FERROFLUID FLOW MODEL.

This article presents a mass-conserved Allen-Cahn type two-phase ferrofluid flow model and establishes its corresponding energy law. The model is a highly coupled, nonlinear saddle point system consisting of the mass-conserved Allen-Cahn equation, the Navier-Stokes equation, the magnetostatic equation, and the magnetization equation. We develop a unified framework of the scalar auxiliary variable (SAV) method and the zero energy contribution (ZEC) approach, which constructs a mass-conserved, fully decoupled, second-order accurate in time, and unconditionally energy-stable linear scheme. We incorporate several distinct numerical techniques, including refor- mulations of the equations to remove the linear couplings and implicit nonlocal integration, the projection method to decouple the velocity and pressure, a symmetric implicit-explicit format for symmetric positive definite nonlinearity, and the continuous finite element method discretization. We also analyze the mass-conserved property, unconditional energy stability, and well-posedness of the scheme. To demonstrate the effectiveness, stability, and accuracy of the developed model and numerical algorithm, we implemented several numerical examples, involving a ferrofluid hedgehog in 2D and a ferromagnetic droplet in 3D. It is worth mentioning that the proposed unified framework of the SAV-ZEC method is also applicable to designing efficient schemes for other coupled-type fluid flow phase-field systems. [ABSTRACT FROM AUTHOR]