In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as applied to gradient flows, and we dissect the mechanisms that enable this method to uphold linearity and the conservation of energy simultaneously. Building upon this foundation, we propose two methods: Invariant Energy Convexification and Invariant Energy Functionalization. These approaches can be perceived as natural extensions of the IEQ method. Employing our novel approaches, we reformulate the system connected to gradient flow, construct a semi-discretized numerical scheme, and obtain a commensurate modified energy dissipation law for both proposed methods. Finally, to underscore their practical utility, we provide numerical evidence demonstrating these methods' accuracy, stability, and effectiveness when applied to both Allen-Cahn and Cahn-Hilliard equations.

翻译：本文深入探究了不变能量二次化（IEQ）方法在梯度流应用中的内在结构复杂性，剖析了该方法同时维持线性性与能量守恒的作用机制。在此基础上，我们提出了两种方法：不变能量凸化法与不变能量泛函化法。这些方法可视为IEQ方法的自然拓展。运用我们的新方法，我们重构了与梯度流相关的系统，构建了半离散数值格式，并为所提两种方法推导出相应的修正能量耗散律。最后，为突出其实用价值，我们提供了数值证据，展示了这些方法在应用于Allen-Cahn和Cahn-Hilliard方程时的精度、稳定性与有效性。