We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward Euler formulas and Crank-Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, meanwhile the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.

翻译：本文提出一种能量优化的不变能量二次化方法，用于求解梯度流模型，该方法仅需在每个时间步上通过一个线性能量优化步骤即可修正辅助变量。除继承了基准方法与松弛不变能量二次化方法的优点外，我们的方法还具有其他若干优势。首先，在修正辅助变量的过程中，我们可通过能量优化技术直接求解线性规划问题，这极大地简化了先前松弛不变能量二次化方法中的非线性优化问题。其次，通过应用向后欧拉公式和克兰克-尼科尔森公式，我们构建了新的线性无条件能量稳定格式，使时间精度可达一阶和二阶。第三，与松弛技术相比，通过能量优化技术获得的修正能量更接近原始能量，同时数值解的准确性和一致性也能得到提升。我们提供了丰富的数值算例，以证明所提格式的准确性、高效性和能量稳定性。