Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy

Abstract

Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are second-order accurate, easy to implement, and maintain the stability of first-order stabilized schemes. We apply the new approach to three different types of gradient flows with strong anisotropic free energy: anisotropic diffusion equation, anisotropic Cahn-Hilliard equation, and Cahn-Hilliard equation with degenerate diffusion mobility. Numerical results are presented to show that the stabilized predictor-corrector schemes are second-order accurate, unconditionally stable for the first two equations, and allow larger time step than the first-order stabilized scheme for the last equation. We also prove rigorously that, for the isotropic Cahn-Hilliard equation, the stabilized predictor-corrector scheme is of second-order.