A generalized SAV approach with relaxation for dissipative systems
A generalized SAV approach with relaxation for dissipative systems
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DOI：
10.1016/j.jcp.2022.111311
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
Yanrong Zhang;Jie Shen
中科院分区：
文献类型：
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作者：
Yanrong Zhang;Jie Shen
标量辅助变量（SAV）方法[31]及其在[20]中提出的推广版本GSAV是构造非线性耗散系统的有效和精确的能量稳定格式的非常流行的方法。然而，SAV的离散值并不直接与耗散系统的自由能相关，如果时间步长不够小，则可能导致不准确的解。受文献[21]中提出的求解梯度流的松弛SAV方法的启发，本文提出了求解一般耗散系统的广义松弛SAV方法（R-GSAV）。R-GSAV方法保留了GSAV方法的所有优点，此外，它耗散了与原始自由能直接相关的修改后的能量。证明了基于R-GSAV的k阶隐式-显式（IMEX）格式是无条件能量稳定的，并对k= 1，2，3，4，5进行了严格的误差分析.我们提出了大量的数值结果来证明改进的精度和有效性的R-GSAV方法。
The scalar auxiliary variable (SAV) approach [31] and its generalized version GSAV proposed in [20] 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 [21] 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 approach, 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.