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 acknowledging 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, 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.

这封信重新介绍了能量平方（EQ）方法，方法是引入一种新颖且必不可少的松弛技术以提高其准确性和一致性。在过去的几年中，EQ方法已得到广泛应用。尽管承认其在为热力学一致的模型设计能量稳定方案方面的有效性，但主要的已知缺点是显而易见的，即，它遵守了由辅助变量而不是原始变量表示的“修改的”能量定律。在数值计算过程中引入了截断误差，因此辅助变量的数值解不再等于其原始连续定义。即使遵守“修改的”能量耗散法则，也无法保证原始的能量耗散法则。在这封信中，我们通过引入松弛步骤来克服这种不一致，为此我们将其命名为relaxed-EQ（REQ）方法。与基线均衡器方法相比，此松弛步骤的计算成本可忽略不计。同时，REQ方法具有所有基线EQ方法的良好特性，例如线性和无条件的能量稳定性。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。