A Revisit of The Energy Quadratization Method with A Relaxation Technique
A Revisit of The Energy Quadratization Method with A Relaxation Technique
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DOI：
10.1016/j.aml.2021.107331
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发表时间：
2021-03
期刊：
影响因子：
--
通讯作者：
Jia Zhao
中科院分区：
文献类型：
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作者：
Jia Zhao
这封信通过引入一种新的和必要的松弛技术来重新审视能量二次化（EQ）方法，以提高其准确性和一致性。在过去的几年里，情商方法已经非常流行了。虽然我们承认它在设计热力学一致模型的能量稳定方案方面的有效性，但已知的主要缺点是明显的，即它尊重由辅助变量而不是原始变量表示的“修改”能量定律。在数值计算过程中引入截断误差，使得辅助变量的数值解不再等同于它们原来的连续定义。数值解虽然尊重“修正”后的能量耗散规律，但不能保证原有的能量耗散规律。在这封信中，我们通过引入松弛步骤来克服这种不一致，我们将其命名为松弛eq （REQ）方法。与基线EQ方法相比，该松弛步骤的计算成本可以忽略不计。同时，REQ方法具有基线EQ方法的所有优点，如线性和无条件能量稳定性。然后，我们将REQ方法应用于几种广泛使用的相场模型，以突出其有效性。
This letter revisits the energy quadratization (EQ) method by introducing a novel and essential relaxation technique to improve its accuracy and consistency. The EQ method has witnessed significant popularity in the past few years. Though we acknowledge its effectiveness in designing energy-stable schemes for thermodynamically consistent models, the primary known drawback is apparent, i.e., it respects a “modified” energy law represented by auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. Even though the “modified” energy dissipation law is respected by the numerical solutions, the original energy dissipation law is not guaranteed. In this letter, we overcome this inconsistency by introducing a relaxation step, for which we named the relaxed-EQ (REQ) method. The computational cost of this relaxation step is negligible compared with the baseline EQ method. Meanwhile, the REQ method holds all the baseline EQ method’s good properties, such as linearity and unconditionally energy stability. Then we apply the REQ method to several widely-used phase field models to highlight its effectiveness.