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.

在本文中，我们为N开发了无条件稳定的线性数值方案组成的Cahn–Hilliard系统，在时空上具有二阶精度。所提出的方案是根据Crank-Nicolson有限差分方案进行修改的，并采用了稳定方法的思想。带有Gauss-Seidel型迭代的非线性多重网格算法用于求解所得离散系统。我们从理论上证明了该方案对于整个系统是无条件稳定的。数值解表明，可以使用较大的时间步长，并在时间和空间上获得二阶精度。并且与线性稳定性分析的结果一致。我们研究了四元混合物中三重结和旋节线分解的演化。此外，可以对提出的方案进行修改，以解决复杂区域和多组分流体流动中的二元旋节线分解。