A Remark on the Invariant Energy Quadratization (IEQ) Method for Preserving the Original Energy Dissipation Laws
A Remark on the Invariant Energy Quadratization (IEQ) Method for Preserving the Original Energy Dissipation Laws
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DOI：
10.3934/era.2022037
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发表时间：
2021-11
期刊：
影响因子：
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通讯作者：
Zengyan Zhang;Yuezheng Gong;Jia Zhao
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文献类型：
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作者：
Zengyan Zhang;Yuezheng Gong;Jia Zhao
在这封信中，我们重新审视不变能量平方化（IEQ）方法，并提供了一个新的视角，它能够保持原来的能量耗散规律。IEQ方法已被广泛用于相场或梯度流模型的能量稳定数值格式设计。虽然IEQ方法有许多优点，但一个主要缺点是IEQ方法通常遵循修正的能量定律，其中修正的能量用辅助变量表示。然而，IEQ方法并不能保证耗散定律在原始能量方面的正确性。使用广泛使用的Cahn-Hilliard方程作为一个例子，我们证明了Runge-Kutta IEQ方法确实可以保持原来的能量耗散规律的某些情况下，任意高阶精度。鼓励感兴趣的读者将这个想法扩展到更一般的情况，并将其应用到其他物理上一致的模型。
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.