Energy-stable method for the Cahn–Hilliard equation in arbitrary domains

Abstract

Phase transition in an irregular domain is a common phenomenon in the natural world. In this study, we develop novel linear, temporally first- and second-order accurate, and unconditionally energy-stable methods for the Cahn–Hilliard (CH) equation in arbitrary domains. We consider a three-component CH system and fix one component as the complex domain. The other two components are updated to simulate the CH dynamics. The contact angle boundary conditions at the interface among three components are achieved by solving the modified version of the CH equation. The scalar auxiliary variable approach transforms the governing equations into equivalent forms. The implicit Euler method and second-order backward difference formula (BDF2) are used to construct time-marching schemes. The main merits of this approach are as follows: (i) The contact angle condition is implicitly achieved by solving a modified ternary CH model. (ii) The proposed schemes are highly efficient because the nonlinear terms are explicitly treated. (iii) The time-discretized energy dissipation laws can be proven analytically. (iv) The implementation in each time step is easy to follow. Numerical experiments indicate that the proposed schemes demonstrate accuracy, energy stability, and superior performance on the CH dynamics in various domains with complex shapes.