An unconditionally stable second-order accurate method for systems of Cahn–Hilliard equations

In this paper, we develop an unconditionally stable linear numerical scheme for the N-component Cahn–Hilliard system with second-order accuracy in time and space. The proposed scheme is modified from the Crank–Nicolson finite difference scheme and adopts the idea of a stabilized method. Nonlinear multigird algorithm with Gauss–Seidel-type iteration is used to solve the resulting discrete system. We theoretically prove that the proposed scheme is unconditionally stable for the whole system. The numerical solutions show that the larger time steps can be used and the second-order accuracy is obtained in time and space; and they are consistent with the results of linear stability analysis. We investigate the evolutions of triple junction and spinodal decomposition in a quaternary mixture. Moreover, the proposed scheme can be modified to solve the binary spinodal decomposition in complex domains and multi-component fluid flows.