An Efficient Numerical Method Of The Viscous Cahn-Hilliard Equation And The Cahn-Hilliard Equation

The Cahn-Hilliard equation is a very important fourth-order nonlinear diffusion equation,which is often used to describe the phase separation and coarsening phenomenon of binary alloys when they are quenched to a certain unstable state.The Cahn-Hilliard equation contains diffusion term and nonlinear term,which makes numerical calculation very difficult.The viscous Cahn-Hilliard equation,arised from the dynamics model,describes the viscous first-order phase transition that occurs when two mixed solutions,such as alloys,are cooled.In this paper,we mainly use the mixed finite element method to solve the viscous Cahn-Hilliard equation.Meanwhile,we propose a second-order accurate in time and energy-stable numerical scheme.In order to construct a second order implicit scheme that satisfied the energy law,we used the modified Crank-Nicolson scheme in time and the mixed finite element method in space.We prove that the proposed method is energy stable and give the error analysis.Due to the nonlinearity and small parameters of the cahn-hilliard equation,a large time step numerical method is proposed in this paper.The main idea is to use conforming finite element method in space,semi-implicit scheme in time,the nonlinear second-order term is treated by convex splitting method and the fourth-order term is solved by implicit scheme.It is found that this method can enlarge the time step.Finally,Numerical experiments are done to demonstrate the effectiveness of our proposed two schemes.