The First Order Numerical Schemes For The Cahn-Hilliard-Navier-Stokes System

As one of the main tools to study interface phenomena,phase field or diffusion interface model has been successfully applied to simulate dynamic processes in many fields.In this paper,the Cahn-Hilliard-Navier-Stokes(CH-NS)phase field model for two-phase incompressible flow is studied.It is a great challenge to establish an effective numerical scheme to solve the coupled phase field model.An important objective of the numerical scheme is to maintain the energy dissipation law at the discrete level.For CH-NS phase field model,there are many research results in recent years.In order to design a simple,efficient,energy stable and discrete energy dissipation law numerical scheme,in this paper,two linear energy stable numerical schemes are proposed,the CH-NS system with polynomial potential function is studied as follows.In the first part,the coupled CH-NS system is studied.In this part,a linear and unconditionally energy stable numerical scheme is proposed.By introducing Lagrange multiplier method to Cahn-Hilliard equation and using pressure corrected projection scheme to Navier-Stokes equation,the whole scheme is linearized.The energy stability of the scheme is proved and the corresponding error analysis is carried out.Finally,a numerical example is given to verify the theoretical analysis.In the second part,the fully decoupled CH-NS system is studied and analyzed.In this part,a linear,unconditional energy stable time discrete scheme is proposed to solve the CH-NS phase field model of two-phase incompressible flow.Based on Lagrange multiplier method and pressure-correction projection scheme,our proposed scheme is linearized by using implicit-explicit treatments,and the calculation of velocity field u,pressure p,phase field variable?are decoupled,so that only linear elliptic equation needs to be solved in each step.By using mathematical induction method,the bound??~n?_L?is obtained.In the framework of the finite element method,the error analysis of variables u,p and?is carried out,and the effectiveness of the method is verified by numerical experiments.