A revisit of the energy quadratization method with a relaxation technique

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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.