Energy stable auxiliary variable method for Cahn--Hilliard equations

In this paper, we propose a quadratic reformulation theory for rational-like functions. Based on this theory, we develop the Quadratic Conservation Elevation (QCE) method, which combines the Scalar Auxiliary Variable (SAV) method with the implicit midpoint rule. We apply this approach to the Cahn-Hilliard (CH) equation with rational-like free-energy terms, obtaining numerical discretizations that preserve the original energy dissipation law. We further derive the discrete dispersion relation and coarsening dynamics, confirming the efficiency and consistency of the method with the continuous counterpart. In addition, we use the proposed method to capture missing orientations for different anisotropic functions. Numerical simulations with various initial conditions illustrate phase separation and anisotropic evolution.