A Remark on the Invariant Energy Quadratization (IEQ) Method for Preserving the Original Energy Dissipation Laws

TL;DR

This paper revisits the IEQ method, showing it can preserve the original energy dissipation laws in phase-field models like the Cahn-Hilliard equation, especially with high-order Runge-Kutta schemes.

Contribution

It demonstrates that the IEQ method can preserve the original energy dissipation laws, correcting the common belief that it only respects a modified energy law.

Findings

IEQ method can preserve original energy laws in certain cases

High-order Runge-Kutta IEQ schemes maintain energy dissipation

Applicable to phase-field and gradient flow models

Abstract

In this letter, we revisit the IEQ method and provide a new perspective on its ability to preserve the original energy dissipation laws. The invariant energy quadratization (IEQ) method has been widely used to design energy stable numerical schemes for phase-field or gradient flow models. Although there are many merits of the IEQ method, one major disadvantage is that the IEQ method usually respects a modified energy law, where the modified energy is expressed in the auxiliary variables. Still, the dissipation laws in terms of the original energy are not guaranteed. Using the widely-used Cahn-Hilliard equation as an example, we demonstrate that the Runge-Kutta IEQ method indeed can preserve the original energy dissipation laws for certain situations up to arbitrary high-order accuracy. Interested readers are highly encouraged to apply our idea to other phase-field equations or gradient flow models.