Abstract
Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights, they exhibit distinct numerical dissipation in smooth regions. This study presents an extension of our previous research which confirmed that adaptively adjusting parameter C can indeed overcome the inadequacy of the usage of a constant small value. Cmin is applied near a discontinuity while Cmax is used elsewhere and they are switched according to the variation of the local flow-field property. This study provides the reference values of the adaptive parameter C of WENOCU4 and systematically evaluates the comprehensive performance of three different switches (labeled as the binary, continuous, and hyperbolic tangent switches, respectively) based on an optimized efficient WENOCU4 scheme (labeled as EWENOCU4). Varieties of 1D scalar equations, empirical dispersion relation analysis, and multi-dimensional benchmark cases of Euler equations are analyzed. Generally, the dissipation and dispersion properties of these three switches are similar. Especially, employing the binary switch, EWENOCU4 achieves the best comprehensive properties. Specifically, the binary switch can efficiently filter more misidentifications in smooth regions than others do, particularly for the cases of 1D scalar equations and Euler equations. Also, the computational efficiency of the binary switch is superior to that of the hyperbolic tangent switch. Moreover, the optimized scheme exhibits high-resolution spectral properties in the wavenumber space. Therefore, employing the binary switch is a more cost-effective improvement for schemes and is particularly suitable for the simulation of complex shock/turbulence interaction. This study provides useful guidance for the reference values of parameter C and the evaluation of adaptive switches.
Abstract
目的
现有研究尚未提供WENOCU4 格式中参数C 的 建议值。本文旨在根据流场特性自适应地调节参 数C 的量级从而提高格式的数值表现。为此,通 过广泛的数值模拟以提供自适应参数C 的参考 值,并系统地评估三种能够自适应调节参数C 的 开关的性能,测得综合表现最佳的开关,进而为 获得高阶WENO 改进型格式提供参考。
创新点
1. 提供了高效WENOCU4 格式的自适应参数C 的参考值;2. 系统地评估了三种自适应开关(二 进制型、连续型和双曲正切型)的性能,并证实 了二进制型开关的最佳表现。
方法
1. 通过理论分析,系统研究三种自适应开关的原 理和性能特点;2. 通过广泛的数值模拟(包括一 维标量方程、经验色散关系和多维欧拉方程的标 准算例),获得自适应参数C 的参考值并验证其 合理性;3. 通过广泛的数值模拟,系统评估三种 自适应开关的综合表现(包括数值色散和耗散特 性以及计算效率),并获得综合性能最佳的自适 应开关。
结论
1. 对于高效WENOCU4 格式而言,本研究证实了 Cmin=40 和Cmax=400 是合理的自适应参数C 的参 考值,因此不应该直接采用WENOCU6 的原始建 议值。2. 根据流场的连续性,采用自适应的参数 C 可以在保证数值稳定性的同时,有效地抑制 WENOCU4 的数值耗散。3. 相比于其它开关,二 进制型开关的综合表现最佳;其能够过滤激波感 知器在光滑区域的一些误判,构造简单,且计算 效率较高。4. 本研究对三种自适应开关的评估具 有一般性,因此易于拓展到其它高阶WENO 格式 的改进工作中。
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Liang LI and Guo-yan ZHAO designed the methodology conception and realized the programming. Da-peng XIONG and Tao TANG processed the corresponding data. Liang LI, Guo-yan ZHAO, and Hong-bo WANG performed formal analysis and investigation. Liang LI wrote the first draft of the manuscript. Guo-yan ZHAO, Hong-bo WANG, and Ming-bo SUN revised and edited the final version.
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Liang LI, Hong-bo WANG, Guo-yan ZHAO, Ming-bo SUN, Da-peng XIONG, and Tao TANG declare that they have no conflict of interest.
Project supported by the National Natural Science Foundation of China (Nos. 11522222, 11925207, and 11472305), the Scientific Research Plan of National University of Defense Technology in 2019 (No. ZK19-02), and the Postgraduate Scientific Research Innovation Project of Hunan Province (Nos. CX20200008 and CX20200084), China
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Li, L., Wang, Hb., Zhao, Gy. et al. Efficient WENOCU4 scheme with three different adaptive switches. J. Zhejiang Univ. Sci. A 21, 695–720 (2020). https://doi.org/10.1631/jzus.A2000006
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DOI: https://doi.org/10.1631/jzus.A2000006