In recent years, the scalar auxiliary variable (SAV) approach has become very popular and hot in the design of linear, high-order and unconditional energy stable schemes of gradient flow models. However, the nature of SAV-based numerical schemes preserving modified energy dissipation limits its wider application. A relaxation technique to correct the modified energy for the baseline SAV method (RSAV) was proposed by Zhao et al. and Shen et al.. The RSAV approach is unconditionally energy stable with respect to a modified energy that is closer to the original free energy, and provides a much improved accuracy when compared with the SAV approach. In this paper, inspired by the RSAV approach, we propose a novel technique to correct the modified energy of the SAV approach, which can be proved to be an optimal energy approximation. We construct new high-order implicit-explicit schemes based on the proposed energy-optimal SAV (EOP-SAV) approach. The constructed EOP-SAV schemes not only provide an improved accuracy but also simplify calculation, and can be viewed as the optimal relaxation. We also prove that the numerical schemes based on the EOP-SAV approach are unconditionally energy stable. Compared with the RSAV approach, the proposed EOP-SAV approach does not need introduce any relaxed factors and can share the similar procedure for error estimates. Several interesting numerical examples have been presented to demonstrate the accuracy and effectiveness of the proposed methods.

翻译：近年来，标量辅助变量（SAV）方法在梯度流模型的线性、高阶及无条件能量稳定格式设计中已变得非常流行且备受关注。然而，基于SAV的数值格式所固有的修正能量耗散性质限制了其更广泛的应用。Zhao等人与Shen等人提出了一种松弛技术，用于修正基础SAV方法（RSAV）的修正能量。RSAV方法在更接近原始自由能的修正能量意义上具有无条件能量稳定性，并且与SAV方法相比，显著提高了计算精度。受RSAV方法启发，本文提出了一种修正SAV方法修正能量的新技术，该技术可被证明是最优能量逼近。我们基于所提出的能量最优SAV（EOP-SAV）方法，构建了新的高阶隐式-显式格式。所构建的EOP-SAV格式不仅提高了计算精度，还简化了计算过程，可视为一种最优松弛方法。我们还证明了基于EOP-SAV方法的数值格式具有无条件能量稳定性。与RSAV方法相比，所提出的EOP-SAV方法无需引入任何松弛因子，且可沿用类似的误差估计流程。本文通过若干数值算例展示了所提方法的精度与有效性。