Energy SSP-IMEX Runge–Kutta methods for the Cahn–Hilliard equation

We consider the Cahn–Hilliard equation, which describes phase separation phenomenon. The equation is discretized by using a fourth-order compact difference scheme in space and strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta methods in time. The new methods have two distinct features: (1) the large time steps can be used in the numerical simulation because the energy is stable, and (2) the energy functional decreases by time. Unconditional energy-stability of first-order and second-order methods are proved. Numerical experiments are given to demonstrate the performance of the proposed methods.