{"title":"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","authors":"Xi Deng, Zhen-hua Jiang, Bin Xie, 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":"{\"title\":\"A Framework Unifying Three-Cell-Based Scale-Invariant Exponential and Trigonometric WENO Weighting Functions With Optimal Shape Parameters\",\"authors\":\"Xi Deng, Zhen-hua Jiang, Bin Xie, 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}","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}
A Framework Unifying Three-Cell-Based Scale-Invariant Exponential and Trigonometric WENO Weighting Functions With Optimal Shape Parameters
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.
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.