SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 143-172, January 2021.
The aim of this study is to develop a novel WENO scheme that improves the performance of the well-known fifth-order WENO methods. The approximation space consists of exponential polynomials with a tension parameter that may be optimized to fit the the specific feature of the data, yielding better results compared to the polynomial approximation space. However, finding an optimal tension parameter is a very important and difficult problem, indeed a topic of active research. In this regard, this study introduces a practical approach to determine an optimal tension parameter by taking into account the relationship between the tension parameter and the accuracy of the exponential polynomial interpolation under the setting of the fifth-order WENO scheme. As a result, the proposed WENO scheme attains an improved order of accuracy (that is, sixth-order) better than other fifth-order WENO methods without loss of accuracy at critical points. A detailed analysis is provided to verify the improved convergence rate. Further, we present modified nonlinear weights based on an $L^1$-norm approach along with a new global smoothness indicator. The proposed nonlinear weights reduce numerical dissipation significantly, while attaining better resolution in smooth regions. Some experimental results for various benchmark test problems are presented to demonstrate the ability of the new scheme.

中文翻译：

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利用指数近似空间提高五阶WENO方案的精度

SIAM数值分析学报，第59卷，第1期，第143-172页，2021年1月。
这项研究的目的是开发一种新型的WENO方案，以提高众所周知的五阶WENO方法的性能。近似空间由具有张力参数的指数多项式组成，可以对多项式进行优化以适合数据的特定特征，与多项式近似空间相比，可获得更好的结果。然而，寻找最佳的张力参数是一个非常重要且困难的问题，这确实是积极研究的主题。在这方面，本研究介绍了一种实用的方法，该方法通过考虑张力参数和五阶WENO方案设置下指数多项式插值精度之间的关系来确定最佳张力参数。结果，提出的WENO方案的准确性得到了提高（即，六阶）优于其他五阶WENO方法，而不会在关键点处损失准确性。提供了详细的分析以验证提高的收敛速度。此外，我们提出了基于$ L ^ 1 $ -norm方法的改进的非线性权重以及新的全局平滑度指标。所提出的非线性权重显着减少了数值耗散，同时在平滑区域中获得了更好的分辨率。提出了一些针对各种基准测试问题的实验结果，以证明新方案的能力。所提出的非线性权重显着减少了数值耗散，同时在平滑区域中获得了更好的分辨率。提出了一些针对各种基准测试问题的实验结果，以证明新方案的能力。所提出的非线性权重显着减少了数值耗散，同时在平滑区域中获得了更好的分辨率。提出了一些针对各种基准测试问题的实验结果，以证明新方案的能力。