The Second Order Dissipative Difference Scheme For The CAHN-ALLEN Equation With NEUMANN Boundary Condition

Abstract In this work, the Neumann initial-boundary value problem of one and two dimensional Cahn-Allen equation are considered. A new semi-implicit dissipative fin- ite difference scheme is proposed and extended to the Cahn-Hilliard. Specifically, for the one and two dimensional Cahn-Allen equation, we propose a second order format in the space direction. Then we prove the existence, stability as well as uniqueness of the numerical solution through the Brower fixed-point theorem, energy method and p- redicted-estimation method. Also, we prove the convergence of the numerical solution and obtain the optimal error estimation. Then,the numerical simulations are performed to demonstrate the effectiveness of the proposed schemes. Furthermore, we extend the scheme to the one dimensional Cahn-Hilliard equation, at the same time, a fourth ord- er format is proposed for the one dimensional Cahn-Allen equation.