论文标题
通用的SAV方法，放松耗散系统
A generalized SAV approach with relaxation for dissipative systems
论文作者
论文摘要
标量辅助变量（SAV）方法\ cite {Shen2018scalar}及其在\ cite {Huang2020highly}中提出的广泛版GSAV是构建非线性散发性系统的有效且准确的能量稳定方案的非常流行的方法。但是，SAV的离散值与耗散系统的自由能直接相关，如果时间步长不够小，可能会导致解决方案不准确。受\ cite {jiang20222improving}的宽松SAV方法的启发，我们在本文中提出了一种通用的SAV方法，并为一般耗散系统放松（R-GSAV）。 R-GSAV方法保留了GSAV评估的所有优势，此外，它消除了与原始自由能直接相关的修改能量。我们证明，基于R-GSAV的$ k $ ther订单隐式解释（IMEX）方案是无条件的能量稳定的，并且我们对$ k = 1,2,3,4,5 $进行了严格的错误分析。我们提出了足够的数值结果，以证明R-GSAV方法的准确性和有效性提高。
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.
