The Dissipative Spectral Scheme For Cahn-Allen Equation And Cahn-Hilliard Equation With Neumann Boundary Condition

In this paper, we focused on the initial-boundary value problem of both Cahn-Allen equation and Cahn-Hilliard equation with Neumann Boundary condition in one dimension as well as in two dimensions. And we delivered an semi-implicit fully discrete dissipative spectral scheme. In detail, we established a spectral scheme with precision of second-order in space direction. The existence of the numerical solution is proved by a series of a priori estimations and the Brower fixed point theorem. The uniqueness of the numerical solution is discussed. The optimal converge rate is obtained by the energy method. The numerical simulations are performed to demonstrate the effectiveness of the proposed schemes.