A generalized SAV approach with relaxation for dissipative systems

Abstract: 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.