The scalar auxiliary variable (SAV) approach \cite{shen2018scalar} and its generalized version GSAV proposed in \cite{huang2020highly} are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in \cite{jiang2022improving} for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV appraoch, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the $k$-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for $k=1,2,3,4,5$. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.
翻译：在\cite{huang2020highly}中提议的Scalar辅助变量(SAV)方法(SAV)及其通用版GSAV(GSAV)方法(在cite{huang2020highly}中提议)是建立非线性散射系统高效和准确能源稳定计划非常流行的方法。然而,SAV的离散价值与消散系统的自由能源没有直接联系,如果时间步骤不够小,可能导致不准确的解决方案。在\cite{jiang2022iSAV(在\cite{jiang2022i)中提议的放松的SAV(R-GSAV)方法的启发下,我们在本文件中提议对一般散射系统采用宽放的通用SAV(R-GSAV)方法(R-GS-GSAV)方法(R-GS-GS)进行严格的错误分析。RAVA1=2,3,4,我们目前对RAV(R)1,3,4,5的精确度进行了精确度分析。
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