Algebraic Invariant Quadratization Schemes for Cahn--Hilliard Equations

In this paper, we propose the Algebraic Invariant Quadratization (AIQ) framework for rational-like energy functions by introducing auxiliary variables, which are interpreted as Casimir functions of the extended system. Combining AIQ with symplectic Runge--Kutta (SRK) methods in time and Fourier pseudo-spectral discretization in space, we obtain fully discrete schemes. The resulting schemes are applied to Cahn--Hilliard equations in both the isotropic and anisotropic cases. We analyze the discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena. Numerical comparisons demonstrate the improved performance superiority of the proposed method over the stabilized invariant energy quadratization (S-IEQ) and scalar auxiliary variable (SAV) methods in preserving the original energy evolution and capturing the underlying physical phenomena.