The Second Order (In Time) Numerical Methods For The Viscous Cahn-Hilliard With Constant And Variable Coefficients

The viscous Cahn-Hilliard equation mainly describes the viscous phase transition when two solutions are cooled such as alloys,glasses and polymer mixtures.The viscous Cahn Hilliard equation is a fourth-order nonlinear equation,and it is difficult to obtain the exact solution in theory.Therefore,it is very important to propose an efficient and energy stable high-order(time)numerical method.The main content of this paper is divided into two parts.For the viscous CahnHilliard equation with constant coefficient and variable coefficient,second-order accurate and unconditionally energy stable numerical schemes are proposed to solve the viscous Cahn-Hilliard equation.In the first part,the mixed finite element method is used to solve the constant coefficient viscous Cahn-Hilliard equation in space.In order to construct a stable second-order semi-implicit scheme,we use Crank-Nicolson/Adams-Bashforth scheme in time.Specifically,the linear term is treated implicitly and the nonlinear term is treated explicitly,which avoids the iterative solution brought by the nonlinear scheme,and a stabilization term is introduced to ensure energy stability.It is proved that the proposed method is energy stable,error analysis and numerical simulation are given.In the second part,the finite element method of viscous Cahn-Hilliard with variable coefficient and the Flory-Huggins potential function is studied.In this paper,a second order(in time)unconditionally energy stable semi-implicit scheme is proposed to solve the equation.The time discretization is based on the Crank-Nicolson scheme and the finite element method is used in space.Where,the nonlinear term are treated explicitly,and two second-order stabilization terms are add to achieve unconditionally energy stability.Because of the complexity of the variable coefficient,the same treatment method as the nonlinear term is adopted,and the linear equations with constant coefficients are easy to be solved numerically.Finally,the stability of method are provided in detail,and the error estimates are given,a series of numerical experiments are implemented to illustrate the theoretical analysis.