We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. To overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. In addition, an optimal rate convergence analysis in the \(\ell ^\infty (0,T; H_N^{-1}) \cap \ell ^2 (0,T; H_N^1)\) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. To deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. A second order accurate, modified BDF scheme is also discussed. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

我们提出并分析了具有三项式Cahn-Hilliard系统的唯一可解且无条件能量稳定的数值方案，具有多项式模式非线性自由能展开。一个关键的困难与三个质量成分的存在有关，尽管总质量约束将其减少为两个成分。另一个数值挑战是确保混合产品形式中的非线性能量泛函的能量稳定性，该非线性能量泛函在三相空间中证明是非凸，非凹的。为了克服这个微妙的困难，我们添加了一些辅助项以使组合的能量在三相空间中凸出，这又导致了三元系统中物理能的凸凹分解。所以，所提出的数值方案的唯一可溶性和无条件的能量稳定性都在理论上建立。此外，在提供了\（\ ell ^ \ infty（0，T; H_N ^ {-1}）\ cap \ ell ^ 2（0，T; H_N ^ 1）\）范数，在空间中进行了傅立叶伪谱离散化是该领域的第一个此类结果。为了处理每个时间步长的非线性隐式方程，我们应用了有效的预处理最速下降（PSD）算法。还讨论了二阶准确的改进BDF方案。给出了一些数值结果，证实了所提出数值方案的稳定性和准确性。