AI 中文总结
针对Cahn-Hilliard方程，提出代数不变二次化框架，结合辛龙格 - 库塔方法和傅里叶伪谱离散化得全离散格式，应用于各向同性和各向异性情况，分析多种现象，数值比较显示其性能优于其他方法。
AI 中文摘要
本文通过引入辅助变量，为类有理能量函数提出代数不变二次化（AIQ）框架，这些辅助变量被解释为扩展系统的卡西米尔函数。将AIQ与时间上的辛龙格 - 库塔（SRK）方法和空间上的傅里叶伪谱离散化相结合，得到全离散格式。所得格式应用于各向同性和各向异性情况下的Cahn-Hilliard方程。分析了离散色散关系、旋节线不稳定性、粗化行为和取向缺失现象。数值比较表明，该方法在保持原始能量演化和捕捉潜在物理现象方面优于稳定不变能量二次化（S-IEQ）和标量辅助变量（SAV）方法。
英文摘要
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.
