  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.
