The scalar auxiliary variable (SAV)-type methods are very popular techniques for solving various nonlinear dissipative systems. Compared to the semi-implicit method, the baseline SAV method can keep a modified energy dissipation law but doubles the computational cost. The general SAV approach does not add additional computation but needs to solve a semi-implicit solution in advance, which may potentially compromise the accuracy and stability. In this paper, we construct a novel first- and second-order unconditional energy stable and positivity-preserving stabilized SAV (PS-SAV) schemes for $L^2$ and $H^{-1}$ gradient flows. The constructed schemes can reduce nearly half computational cost of the baseline SAV method and preserve its accuracy and stability simultaneously. Meanwhile, the introduced auxiliary variable is always positive while the baseline SAV cannot guarantee this positivity-preserving property. Unconditionally energy dissipation laws are derived for the proposed numerical schemes. We also establish a rigorous error analysis of the first-order scheme for the Allen-Cahn type equation in $l^{\infty}(0,T; H^1(\Omega) ) $ norm. In addition we propose an energy optimization technique to optimize the modified energy close to the original energy. Several interesting numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods.

翻译：标量辅助变量（SAV）类方法是求解各类非线性耗散系统的热门技术。与半隐式方法相比，基准SAV方法虽能保持修正能量耗散律，但计算量翻倍。通用SAV方法无需额外计算，但需预先求解半隐式解，可能影响精度与稳定性。本文针对$L^2$和$H^{-1}$梯度流，构建了新型一阶与二阶无条件能量稳定且保持正性的稳定化SAV（PS-SAV）格式。所构造格式可将基准SAV方法的计算成本降低近半，同时保持其精度与稳定性。此外，引入的辅助变量始终为正，而基准SAV无法保证这一正性保持特性。本文推导了所提数值格式的无条件能量耗散律，并建立了Allen-Cahn型方程一阶格式在$l^{\infty}(0,T; H^1(\Omega) )$范数下的严格误差分析。同时提出能量优化技术，使修正能量趋近原始能量。通过若干数值算例验证了所提方法的精度与有效性。