A novel fully decoupled scheme with second‐order time accuracy and unconditional energy stability for the Navier‐Stokes equations coupled with mass‐conserved Allen‐Cahn phase‐field model of two‐phase incompressible flow

摘要

Abstract We consider the numerical approximation of the flow‐coupled phase‐field model of two‐phase incompressible flows using the mass‐conserved Allen‐Cahn equation. Due to the highly nonlinear nature of the coupling, how to develop an accurate and practically efficient scheme has always been a challenging problem. To solve this challenge, we construct a novel effective fully decoupled scheme that is linear, unconditional energy stable, and second‐order time accurate. The key idea of decoupling is to introduce a nonlocal variable and a related ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so‐called “zero‐energy‐contribution” property. Thus, in actual calculations, this scheme only needs to solve several independent linear equations at each time step to obtain a numerical solution with the second‐order time accuracy. We strictly prove the solvability and unconditional energy stability and perform numerical simulations in 2D and 3D to verify the accuracy and stability of the scheme numerically.