A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen–Cahn type flow-coupled binary surfactant model

Abstract

In this paper, we establish a binary fluid surfactant model by coupling two mass-conserved Allen–Cahn equations and the Navier–Stokes equations and consider numerical simulations of the developed model. Due to a large number of nonlinear and nonlocal coupling terms in the model, it is very challenging to design an efficient and accurate numerical scheme, especially the full decoupling scheme with second-order time accuracy. We solve this challenge by developing a novel fully-decoupled approach, where the key idea achieving the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so-called “zero-energy-contribution” property. In this way, we can easily discretize the coupled nonlinear terms in a fully explicit way, while still maintaining unconditional energy stability. By combining with the projection type method and quadratization approach, at each time step, we only need to solve several fully-decoupled linear elliptic equations with constant coefficients. We strictly prove the solvability of the scheme, prove that the scheme satisfies the unconditional energy stability, and give various 2D and 3D numerical simulations to show its stability and accuracy numerically. As far as the author knows, this is the first fully-decoupled and second-order time-accurate scheme of the flow-coupled phase-field type model.