On a Novel Fully Decoupled, Second-Order Accurate Energy Stable Numerical Scheme for a Binary Fluid-Surfactant Phase-Field Model

The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and Navier--Stokes equations, is a very complex nonlinear system, which poses many challenges to the design of numerical schemes. As far as the author knows, due to the highly nonlinear coupling nature, there is no fully decoupled scheme with second-order accuracy in time for numerical approximation. This paper proposes a novel decoupling approach by introducing a nonlocal auxiliary variable and its associated ODE to deal with the nonlinear coupling terms that satisfy the so-called zero-energy-contribution property. By combining it with other proven effective methods (the projection method of the Navier--Stokes equations and the SAV method of linearizing nonlinear potential), we arrive at a fully decoupled, linear, unconditionally energy stable scheme with second-order time accuracy. At each time step, only a few fully decoupled linear elliptic equations with constant coefficients are needed to be solved, which shows the advantages of ease of implementation and efficiency. We also prove the unconditional energy stability rigorously and provide various numerical simulations in two and three dimensions to demonstrate its stability and accuracy, numerically.