Stabilized semi-implicit numerical schemes for the Cahn-Hilliard-like equation with variable interfacial parameter

We present stabilized semi-implicit schemes for the MMC-TDGL equation, a Cahn-Hilliard-like equation with variable interfacial parameter. By introducing two stabilization terms, we decompose the 4th-order variable coefficient PDE into two decoupled Helmholtz type equations with the constant coefficients. The proposed scheme is proved to be stable, which means the decay of energy is preserved. Due to the periodic boundary condition, the fast Fourier transformation method is applied to accelerate the computations. To improve the stability of the proposed scheme, we present a nonlinear stabilized semi-implicit scheme, which can be simulated efficiently with fixed-point iteration for the nonlinear terms. Some numerical experiments are carried out to simulate the phase transition, to verify the total energy decay, and to observe the influence of some parameters. The strictly energy-stable scheme saves much CPU time due to the constant coefficients( with FFT method) without losing the accuracy. It is the key novelty.