A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws

Zengyan Zhang; Yuezheng Gong; Jia Zhao; Zengyan Zhang; Yuezheng Gong; Jia Zhao

doi:10.3934/era.2022037

Electronic Research Archive

2022, Volume 30, Issue 2: 701-714. doi: 10.3934/era.2022037

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A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws

1.
Department of Mathematics & Statistics, Utah State University, Logan, UT 84322, USA
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
3.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

Received: 24 November 2021 Revised: 13 January 2022 Accepted: 13 January 2022 Published: 22 February 2022

In this letter, we revisit the invariant energy quadratization (IEQ) method and provide a new perspective on its ability to preserve the original energy dissipation laws. The 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 by the IEQ method. 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 encouraged to extend this idea to more general cases and apply it to other thermodynamically consistent models.
- energy stable,
- cahn hilliard equation,
- invariant energy quadratization (IEQ) method
Citation: Zengyan Zhang, Yuezheng Gong, Jia Zhao. A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws[J]. Electronic Research Archive, 2022, 30(2): 701-714. doi: 10.3934/era.2022037

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Abstract

In this letter, we revisit the invariant energy quadratization (IEQ) method and provide a new perspective on its ability to preserve the original energy dissipation laws. The 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 by the IEQ method. 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 encouraged to extend this idea to more general cases and apply it to other thermodynamically consistent models.

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