Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

Abstract

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.

Appendix

Here, we briefly describe the first-order time-accurate and fully discrete scheme. Based on a backward Euler formula, we discretize Eqs. (8)–(10) to be

$$\begin{aligned} \frac{c^{n+1}_{k,ij}-c^{n}_{k,ij}}{\Delta t} =& \Delta _{d} \mu ^{n+1}_{k,ij}, \end{aligned}$$

(72)

$$\begin{aligned} \mu ^{n+1}_{k,ij} =& -\epsilon ^{2}\Delta _{d} c^{n+1}_{k,ij} + q^{n+1}_{k,ij}c^{n}_{k,ij} - \frac{1}{2}q^{n+1}_{k,ij} + \beta ({\mathbf {c}}^{n}_{ij}), \end{aligned}$$

(73)

$$\begin{aligned} \frac{q^{n+1}_{k,ij}-q^{n}_{k,ij}}{\Delta t} =& (2c^{n+1}_{k,ij}-1) \frac{c^{n+1}_{k,ij}-c^{n}_{k,ij}}{\Delta t},~~~\mbox{for}~k = 1,2,3. \end{aligned}$$

(74)

The periodic or homogeneous Neumann boundary is considered. The energy stability, mass conservation, and unique solvability of the above scheme can be proved by following same procedures described in Sect. 3. We leave them to interested readers.