A novel efficient generalized energy-optimized exponential SAV scheme with variable-step BDFk method for gradient flows

In this paper, we propose a novel generalized energy-optimized (GEOP) technique to correct the modified energy of the scalar auxiliary variable (SAV) approach for gradient flows. Firstly, we use the variable-step kth-order (k = 2,3) backward differentiation formula (BDFk) to construct a linear exponential SAV method (ESAV), denoted as BDFk-ESAV. This method is shown to preserve only a modified energy dissipation law. To address this issue, we present an energy-optimized (EOP) technique derived from a novel linear relaxation strategy, which penalizes the inconsistency between the SAV and the original nonlinear potential energy. However, this does not always bring the modified energy closer to the original total energy. This paper presents one essential GEOP technique to overcome this issue, which leads to a novel ESAV scheme, namely BDFk-GEOP-ESAV. We demonstrate that this scheme unconditionally satisfies the modified energy dissipation law, similar to the proposed ESAV and EOP-ESAV schemes. Most importantly, its energy is an optimal approximation to the original total energy, not just the nonlinear potential energy. Therefore, it enables a broad range of applications for long-term stable modeling. Additionally, an improved adaptive time-stepping strategy is developed to further enhance the effectiveness and efficiency of the variable-step BDFk-GEOP-ESAV method. Representative numerical examples are presented to illustrate the superior performance of the proposed methods.