Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation
Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation
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DOI：
10.1016/j.jcp.2022.110954
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
Maosheng Jiang;Zengyan Zhang;Jia Zhao
中科院分区：
文献类型：
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作者：
Maosheng Jiang;Zengyan Zhang;Jia Zhao
标量辅助变量（SAV）方法是由Shen等人在[36]中提出的，并已被广泛地用于求解相容偏微分方程问题。利用标量辅助变量，将原偏微分方程问题转化为等价的偏微分方程问题。SAV方法的优点，如线性度，无条件能量稳定性，易于实现，是普遍的。然而，仍然有一个悬而未决的问题，即，由SAV方法得到的数值格式根据辅助变量而不是原始变量来保持“修改的”能量定律。在数值计算过程中引入截断误差，使得辅助变量的数值解不再等价于其原始连续定义。换句话说，即使SAV方案满足修改的能量定律，它不一定满足原始PDE模型的能量定律。本文提出了一种基本的松弛技术来克服这个问题，我们命名为松弛SAV（RSAV）方法。我们的RSAV方法惩罚辅助变量的数值误差的松弛技术。总的来说，RSAV方法保留了基线SAV方法的所有优点，并显著提高了其精度和一致性。已经提出了几个例子来证明RSAV方法的有效性。
The scalar auxiliary variable (SAV) method was introduced by Shen et al. in [36] and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulting from the SAV method preserve a “modified” energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach.