We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

我们为三元Cahn-Hilliard（CH）模型开发了一种二阶准确，能量稳定且线性的数值方法。所提出的方案是对二进制CH系统的典型拉格朗日乘数方法的扩展。应用二阶后向差分公式（BDF2）构造时间离散化。我们从理论上证明了该方案的质量守恒，独特的可溶性和能量稳定性。我们通过使用多重网格算法有效地解决了所得的离散线性系统。数值解证明了该方案在时间和空间上是实用稳定的，并且是二阶精确的。此外，我们可以将所提方案用作三元相场流体系统中三元CH方程的有效求解器。