Linear and conservative IMEX Runge–Kutta finite difference schemes with provable energy stability for the Cahn–Hilliard model in arbitrary domains

The Cahn–Hilliard (CH) equation is a classical mathematical equation which models the spinodal decomposition in binary fluid mixtures. In the real world, the phase change in a two-phase system usually occurs in irregular domains. To efficiently and accurately simulate this dynamics, we herein develop linear temporally first- and second-order accurate methods for the CH equation in arbitrary domains. The implicit-explicit (IMEX) Runge–Kutta method is adopted to construct discrete schemes in time. By introducing a simple boundary control function, we transform the original equations into equivalent forms in irregular domains and discretize the space by using the standard second-order finite difference stencil. In each temporal step, the proposed numerical schemes are highly efficient and easy to implement because we only need to solve several linear elliptic type equations. The mass conservation and unconditional energy stability of the proposed schemes are analytically proved. The multigrid algorithm is adopted for fast computation. The numerical results confirm the desired accuracy in time and space, mass conservation, and energy dissipation property. Moreover, the extensive calculations in two- and three-dimensional spaces indicate that the proposed method has good capability to simulate spinodal decomposition in arbitrarily irregular domains.