A Framework Unifying Three-Cell-Based Scale-Invariant Exponential and Trigonometric WENO Weighting Functions With Optimal Shape Parameters

IF 1.8 4区 工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Xi Deng, Zhen-hua Jiang, Bin Xie, Chao Yan
{"title":"A Framework Unifying Three-Cell-Based Scale-Invariant Exponential and Trigonometric WENO Weighting Functions With Optimal Shape Parameters","authors":"Xi Deng,&nbsp;Zhen-hua Jiang,&nbsp;Bin Xie,&nbsp;Chao Yan","doi":"10.1002/fld.5401","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>Exponential and trigonometric functions have been extensively employed as the kernel of reconstruction operators within numerous WENO (Weighted Essentially Non-oscillatory) schemes to accelerate the convergence rate. However, most of them are scale-dependent, compromising the robustness required for multi-scale flow simulations. Thus, this work aims to develop novel three-cell-based scale-invariant WENO schemes that use exponential and trigonometric functions as the kernel of non-linear weights. First, to achieve the scale-invariant property, this work reformulates the newly proposed scale-invariant ROUND (Reconstruction Operators in Unified Normalized-variable Diagram) schemes into the form of WENO weighting functions, thereby facilitating the design of scale-invariant WENO schemes. Then, this work proposes new WENO non-linear weights using exponential and trigonometric functions—such as Gaussian, hyperbolic, and cosine functions—to enhance the accuracy of the three-cell-based WENO scheme. The proposed WENO weights contain a shape parameter that controls the errors between the non-linear weight and the ideal weight. As the value of the shape parameter increases, the non-linear weight converges towards the ideal weight but also becomes more likely to produce numerical oscillations. To approximate the optimal value of the shape parameter, the WENO reconstruction operator is projected into normalized variable space, and the shape parameters are fine-tuned to ensure the normalized reconstruction operator falls into the CBC (Convection Bounded Criterion) region of UND (Unified Normalized-variable Diagram). The accuracy analysis reveals that the proposed weighting functions outperform classical WENO schemes, particularly when the smooth function contains a first-order critical point. The accuracy and shock-capturing properties of the proposed schemes are further validated through benchmark tests. Thus, this work demonstrates using the ROUND framework to design scale-invariant three-cell-based WENO schemes with exponential and trigonometric functions and optimal shape parameters.</p>\n </div>","PeriodicalId":50348,"journal":{"name":"International Journal for Numerical Methods in Fluids","volume":"97 9","pages":"1209-1225"},"PeriodicalIF":1.8000,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal for Numerical Methods in Fluids","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/fld.5401","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0

Abstract

Exponential and trigonometric functions have been extensively employed as the kernel of reconstruction operators within numerous WENO (Weighted Essentially Non-oscillatory) schemes to accelerate the convergence rate. However, most of them are scale-dependent, compromising the robustness required for multi-scale flow simulations. Thus, this work aims to develop novel three-cell-based scale-invariant WENO schemes that use exponential and trigonometric functions as the kernel of non-linear weights. First, to achieve the scale-invariant property, this work reformulates the newly proposed scale-invariant ROUND (Reconstruction Operators in Unified Normalized-variable Diagram) schemes into the form of WENO weighting functions, thereby facilitating the design of scale-invariant WENO schemes. Then, this work proposes new WENO non-linear weights using exponential and trigonometric functions—such as Gaussian, hyperbolic, and cosine functions—to enhance the accuracy of the three-cell-based WENO scheme. The proposed WENO weights contain a shape parameter that controls the errors between the non-linear weight and the ideal weight. As the value of the shape parameter increases, the non-linear weight converges towards the ideal weight but also becomes more likely to produce numerical oscillations. To approximate the optimal value of the shape parameter, the WENO reconstruction operator is projected into normalized variable space, and the shape parameters are fine-tuned to ensure the normalized reconstruction operator falls into the CBC (Convection Bounded Criterion) region of UND (Unified Normalized-variable Diagram). The accuracy analysis reveals that the proposed weighting functions outperform classical WENO schemes, particularly when the smooth function contains a first-order critical point. The accuracy and shock-capturing properties of the proposed schemes are further validated through benchmark tests. Thus, this work demonstrates using the ROUND framework to design scale-invariant three-cell-based WENO schemes with exponential and trigonometric functions and optimal shape parameters.

一种统一具有最优形状参数的三格尺度不变指数和三角WENO加权函数的框架
指数函数和三角函数作为重构算子的核被广泛应用于WENO(加权本质非振荡)格式中,以加快收敛速度。然而,它们大多是尺度相关的,影响了多尺度流动模拟所需的鲁棒性。因此,这项工作旨在开发新的基于三单元的尺度不变WENO方案,该方案使用指数和三角函数作为非线性权重的核。首先,为了实现尺度不变性,本文将新提出的尺度不变性ROUND(统一归一化变量图重构算子)方案重新表述为WENO加权函数的形式,从而便于WENO方案的尺度不变性设计。然后,本文利用指数函数和三角函数(如高斯函数、双曲函数和余弦函数)提出了新的WENO非线性权重,以提高基于三单元的WENO方案的准确性。所提出的WENO权值包含一个形状参数,用于控制非线性权值与理想权值之间的误差。随着形状参数的增大,非线性权值向理想权值收敛,但也更容易产生数值振荡。为了逼近形状参数的最优值,将WENO重构算子投影到归一化变量空间中,并对形状参数进行微调,使归一化重构算子落在统一归一化变量图(UND)的CBC (Convection Bounded Criterion)区域。精度分析表明,所提出的加权函数优于经典WENO方案,特别是当光滑函数包含一阶临界点时。通过基准测试进一步验证了所提方案的精度和冲击捕获性能。因此,这项工作展示了使用ROUND框架来设计具有指数函数和三角函数以及最优形状参数的尺度不变的基于三单元的WENO方案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
International Journal for Numerical Methods in Fluids
International Journal for Numerical Methods in Fluids 物理-计算机:跨学科应用
CiteScore
3.70
自引率
5.60%
发文量
111
审稿时长
8 months
期刊介绍: The International Journal for Numerical Methods in Fluids publishes refereed papers describing significant developments in computational methods that are applicable to scientific and engineering problems in fluid mechanics, fluid dynamics, micro and bio fluidics, and fluid-structure interaction. Numerical methods for solving ancillary equations, such as transport and advection and diffusion, are also relevant. The Editors encourage contributions in the areas of multi-physics, multi-disciplinary and multi-scale problems involving fluid subsystems, verification and validation, uncertainty quantification, and model reduction. Numerical examples that illustrate the described methods or their accuracy are in general expected. Discussions of papers already in print are also considered. However, papers dealing strictly with applications of existing methods or dealing with areas of research that are not deemed to be cutting edge by the Editors will not be considered for review. The journal publishes full-length papers, which should normally be less than 25 journal pages in length. Two-part papers are discouraged unless considered necessary by the Editors.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信