Algebraic Invariant Quadratization Schemes for Cahn--Hilliard Equations
arXiv:2607.11569
The paper introduces an Algebraic Invariant Quadratization (AIQ) framework with auxiliary Casimir variables, combined with symplectic Runge‑Kutta time stepping and Fourier pseudo‑spectral spatial discretization, to develop energy‑stable schemes for isotropic and anisotropic Cahn–Hilliard equations and demonstrates superior performance over existing IEQ and SAV methods.
Abstract
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.