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

Two novel numerical methods for gradient flows: generalizations of the Invariant Energy Quadratization method