An unconditionally energy stable algorithm for copolymer–homopolymer mixtures

Abstract

The homopolymer and copolymer mixtures have attracted extensive research in various scientific and engineering fields. A highly efficient, temporally second-order accurate, and unconditionally energy stable scheme for the coupled Cahn–Hilliard system simulating the phase separation in the homopolymer and copolymer mixtures is proposed. To appropriately treat the nonlinear and coupling terms in the coupled Cahn–Hilliard system, an efficient variant of the scalar auxiliary variable (SAV) approach is considered. However, the time-discretized modified energy resulting from the SAV method is usually no longer consistent with the original continuous energy because of the truncation errors during numerical implementations. Therefore, a simple and practical energy relaxation technique is adopted to overcome this shortcoming, and then the time-discretized corrected energy is obtained. The present method has the following merits: (i) In each iteration, all variables are decoupled, and we can solve them step-by-step. Thus, the numerical implementation is efficient; (ii) The time-discretized corrected energy and the original energy are highly consistent; (iii) The unique solvability and energy stability are strictly proved; (iv) Various morphologies of pattern formations of the copolymer–homopolymer mixtures can be well simulated. The accuracy, stability, and consistency of developed scheme are demonstrated by various 2D and 3D numerical experiments.