Efficient energy-stable second-order predictor-corrector SAV scheme for the Cahn-Hilliard equation: algorithm, analysis, and computation

The Cahn-Hilliard equation plays a central role in modeling phase separation processes in complex systems, including alloys, polymers, and biological materials. Numerical schemes for this equation must balance efficiency, stability, and accuracy in order to capture the rich dynamics of interfacial evolution. In this work, we developed a new second-order predictor–corrector scheme within the scalar auxiliary variable (SAV) framework, combined with Crank–Nicolson (CN) time discretization. The proposed method is linear, uniquely solvable, and unconditionally energy stable, while also providing rigorous error estimates. Computational experiments demonstrated that the new scheme not only maintains second-order temporal accuracy for relatively large time steps, but also yields smaller numerical errors compared to standard SAV-CN methods. These results highlight both the theoretical advantages and practical potential of the predictor–corrector SAV approach for advancing accurate and efficient simulations of phase-field models in science and engineering.