Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique

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 differentiation 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, and 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.