Comparison Of Several Numerical Methods For 1-D Cahn-hilliard Equation

In this thesis, we consider numerical method for the Cahn-Hilliard equation in one dimension. We develop a direct discontinuous Galerkin(DDG) method for the Cahn-Hilliard equation, and prove that the proposed DDG method is mass preserv-ing and energy dissipative. We numerically compare the DDG method with three finite difference methods including the Crank-Nicolson Adams-Bashforth FDM, the DVDM(discrete variational derivative method)-based FDM and the FDM with Pi-card iteration. Four numerical examples are presented to show the performance of the four numerical methods which are all mass preserving and energy dissipative.When applying the forward Euler scheme for time discretization in the DDG method,it requires smaller time step in comparison with the finite difference method.