Highly efficient variant of SAV approach for two-phase incompressible conservative Allen-Cahn fluids

Herein, we construct efficient linear, totally decoupled, and energy dissipative schemes for two-phase incompressible conservative Allen-Cahn (CAC) fluid model. The binary CAC model has good potential in simulating fluid flow with interface because of the following merits: (i) mass conservation is satisfied, (ii) interfacial position can be easily captured. Comparing with the well-known fourth-order Cahn-Hilliard (CH) equation, the CAC equation is easy to solve since its second-order property. The scalar auxiliary variable (SAV)-type methods provide practical approach to develop linearly energy-stable schemes for phase-field problems. A variant of SAV approach considered in this work not only leads to accurate schemes for CAC fluid system, but also achieves highly efficient calculation. In each time step, only several linear and decoupled equations need to be computed. The linear multigrid algorithm is adopted to accelerate convergence. The unique solvability, modified energy dissipation law, and mass conservation in time-discretized version are analytically proved. Extensive numerical experiments are performed to validate the superior performance of the proposed methods.