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 reformulations 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.Keywordsferrofluidtwo-phasemass-conserved Allen–Cahnenergy stabilitydecouplingmagnetic fieldMSC codes65N1265M1265M70