Review on non-polynomial high-order accuracy nonlinear weighted schemes

Li Haomin; Mao Meiliang; Bai Jinwei; Dong Weizhong; Zhu Huajun

doi:10.7638/kqdlxxb-2025.0070

English Version Journal Login

您当前的位置：

首页 >

文章列表页 >

Review on non-polynomial high-order accuracy nonlinear weighted schemes

更新时间：2026-05-28

- Review on non-polynomial high-order accuracy nonlinear weighted schemes
- Vol. 44, Issue 1, Pages: 6-18(2026)
- 作者机构：
  
  1. 中国空气动力研究与发展中心,绵阳,621000
  2. 空天飞行空气动力科学与技术全国重点实验室,绵阳,621000
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2025.0070
  CLC： V233.1;V211.3
- Published：2026
- 稿件说明：
移动端阅览
黎昊旻, 毛枚良, 白进维, 等. 非多项式高阶非线性加权格式研究综述[J]. 空气动力学学报, 2026, 44(1): 6−18.

Li Haomin, Mao Meiliang, Bai Jinwei, et al. Review on non-polynomial high-order accuracy nonlinear weighted schemes[J]. 2026, 44(1): 6-18.
黎昊旻, 毛枚良, 白进维, 等. 非多项式高阶非线性加权格式研究综述[J]. 空气动力学学报, 2026, 44(1): 6−18. DOI： 10.7638/kqdlxxb-2025.0070

Li Haomin, Mao Meiliang, Bai Jinwei, et al. Review on non-polynomial high-order accuracy nonlinear weighted schemes[J]. 2026, 44(1): 6-18. DOI： 10.7638/kqdlxxb-2025.0070.

摘要

在数值求解双曲守恒律方程弱解时，基于多项式的高精度非线性加权格式与线性格式相比，其引入的非线性误差会对格式分辨率产生不可忽略的影响。为了降低这种影响，学者们一方面在加权模板集选取、光滑因子计算及其与非线性权函数关系等方面开展了系统性的研究工作；另一方面，也对基于非多项式的高精度非线性加权格式进行了探索性研究。本文归纳总结了三角函数、对数函数、径向基函数及双曲正切函数等非多项式的WENO格式研究成果，展示了它们的频谱分辨率和间断捕捉能力等方面的特性。三角函数和径向基函数的WENO格式在频谱分辨率上优于多项式WENO格式，双曲正切函数的WENO格式在捕捉间断时产生的数值耗散明显低于多项式的WENO格式。这些初步而积极的研究成果，反映了WENO格式研究的新动向，对推进高阶精度非线性加权格式研究具有重要指导意义。

Abstract

In comparison to linear schemes

the nonlinear errors inherent in polynomial-based high-order nonlinear weighted schemes significantly affect the resolution when solving weak solutions of hyperbolic conservation laws. To mitigate this impact

researchers have systematically investigated aspects such as weighted stencils

smoothness indicators

and nonlinear weighting functions. Additionally

exploratory studies have been undertaken on high-order nonlinear weighted schemes based on non-polynomial approaches. This paper reviews the advancements in non-polynomial weighted essentially non-oscillatory (WENO) schemes

including trigonometric functions

logarithmic functions

radial basis functions (RBF)

and hyperbolic tangent functions (THINC). Their performance is highlighted in terms of spectral resolution and shock-capturing capabilities. Trigonometric and RBF-based WENO schemes demonstrate superior spectral resolution compared to polynomial-based WENO schemes

while THINC-based WENO schemes exhibit significantly lower numerical dissipation when capturing discontinuities. These preliminary yet promising findings reflect a new trend in WENO scheme research and provide valuable guidance for advancing the study of high-order nonlinear weighted schemes.

关键词

Keywords

references

Cary A W, Chawner J, Duque E P, et al. CFD vision 2030 road map: Progress and perspectives[C]// AIAA Aviation 2021 Forum: American Institute of Aeronautics and Astronautics, 2021. doi: 10.2514/6.2021-2726

Li Xinliang, Fu Dexun, Ma Yanwen. Optimized group velocity control scheme and DNS of decaying compressible turbulence of relative high turbulent Mach number[J]. International Journal for Numerical Methods in Fluids, 2005, 48(8): 835−852. doi: 10.1002/fld.941

Billet G, Louedin O. Adaptive limiters for improving the accuracy of the MUSCL approach for unsteady flows[J]. Journal of Computational Physics, 2001, 170(1): 161−183. doi: 10.1006/jcph.2001.6731

Gottlieb S, Shu C-W. Total variation diminishing Runge-Kutta schemes[J]. Mathematics of Computation, 1998, 67(221): 73−85. doi: 10.1090/s0025-5718-98-00913-2

Zhang Di, Jiang Chunbo, Liang Dongfang, et al. A review on TVD schemes and a refined flux-limiter for steady-state calculations[J]. Journal of Computational Physics, 2015, 302: 114−154. doi: 10.1016/j.jcp.2015.08.042

Zalesak S T. Fully multidimensional flux-corrected transport algorithms for fluids[J]. Journal of Computational Physics, 1979, 31(3): 335−362. doi: 10.1016/0021-9991(79)90051-2

Boris J P, Book D L. Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works[J]. Journal of Computational Physics, 1973, 11(1): 38−69. doi: 10.1016/0021-9991(73)90147-2

Zhu Jun, Shu C-W, Qiu Jianxian. Development and prospect of high-order WENO schemes[J]. Scientia Sinica Mathematica, 2024, 54(2): 121−138 (in Chinese). doi: 10.1360/SSM-2023-0236

Adams N A, Shariff K. A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems[J]. Journal of Computational Physics, 1996, 127(1): 27−51. doi: 10.1006/jcph.1996.0156

Mao Meiliang, Zhu Huajun, Deng Xiaogang, et al. Effect of geometric conservation law on improving spatial accuracy for finite difference schemes on two-dimensional nonsmooth grids[J]. Communications in Computational Physics, 2015, 18(3): 673−706. doi: 10.4208/cicp.250614.060215a

Deng Xiaogang, Maekawa H, Shen C. A class of high-order dissipative compact schemes[C]//Fluid Dynamics Conference, 2013. doi: 10.2514/6.1996-1972

Deng Xiaogang, Jiang Yi, Mao Meiliang, et al. A family of hybrid celledge and cell-node dissipative compact schemes satisfying geometric conservation law[J]. Computers Fluids, 2015, 116: 29−45. doi: 10.1016/j.compfluid.2015.04.015

Lele S. Compact finite difference schemes with spectral-like resolution[J]. Journal of Computational Physics, 1992, 103(1): 16−42. doi: 10.1016/0021-9991(92)90324-R

Tu Guohua, Deng Xiaogang, Mao Meiliang. Spectral property comparison of fifth-order nonlinear WCNS and WENO difference schemes[J]. Acta Aerodynamica Sinica, 2012, 30(6): 709−712(in Chinese).

MAO M L, YAN Z G, LIU H Y, et al. Study of quasi-linear spectral analysis method of high-order weighted nonlinear schemes[J]. Acta Aerodynamica Sinica, 2015, 33(1): 1−9(in Chinese). doi: 10.7638/kqdlxxb-2014.0115

Deng Xiaogang, Mao Meiliang, Jiang Yi, et al. New high-order hybrid cell-edge and cell-node weighted compact nonlinear schemes[C]//20th AIAA Computational Fluid Dynamics Conference, 2011. doi: 10.2514/6.2011-3857

Deng Xiaogang, Mao Meiliang. Weighted compact high-order nonlinear schemes for the Euler equations[C]//13th Computational Fluid Dynamics Conference, 1997. doi: 10.2514/6.1997-1941

Bai Jinwei, Yan Zhenguo, Mao Meiliang, et al. A new fifth-order weighted compact nonlinear scheme with multi-order candidates weighting for hyperbolic conservation laws[J]. International Journal of Computational Fluid Dynamics, 2024, 38(1): 1−27. doi: 10.1080/10618562.2024.2401825

Bai Jinwei, Yan Zhenguo, Mao Meiliang, et al. Improvement of Z-weighted function based on fifth-order nonlinear multi-order weighted method for shock capturing of hyperbolic conservation laws[J]. Entropy, 2024, 26(4). doi: 10.3390/e26040334

Bai Jinwei, Mao Meiliang, Ma Yankai, et al. Simplified weighting formulations of weighted compact nonlinear schemes for compressible flows[J]. International Journal for Numerical Methods in Fluids, 2024, 96(9): 1592−1616. doi: 10.1002/fld.5311

Ma Yankai, Mao Meiliang, Yan Zhenguo, et al. A multi-order candidates weighting framework for discontinuity capturing of hyperbolic conservation laws[J]. Journal of Computational Physics, 2024, 510: 113064. doi: 10.1016/j.jcp.2024.113064

Shu C-W, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1989, 77(2): 439−471.

Shu C-W, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. II[J]. Journal of Computational Physics, 1989, 83(1): 32-78. doi: 10.1016/0021-9991(89)90222-2

Shu C-W. High order weighted essentially nonoscillatory schemes for convection dominated problems[J]. SIAM Review, 2009, 51(1): 82−126. doi: 10.1137/070679065

Kim C H, Ha Y, Yoon J. Modified non-linear weights for fifth order weighted essentially non-oscillatory schemes[J]. Journal of Scientific Computing, 2016, 67(1): 299−323. doi: 10.1007/s10915-015-0079-3

Mao Meiliang, Bai Jinwei, Min Yaobing, et al. Review on weighting functions of high-order nonlinear weighted methods[J]. Acta Aerodynamica Sinica, 2024, 42(6): 1−14(in Chinese). doi: 10.7638/kqdlxxb-2023.0187

邓小刚, 刘昕, 毛枚良, 等. 高精度加权紧致非线性格式的研究进展[J]. 力学进展, 2007, 37(3): 417−427.

Fu Lin. Review of the high-order TENO schemes for compressible gas dynamics and turbulence[J]. Archives of Computational Methods in Engineering, 2023: 2493−2526. doi: 10.1007/s11831-022-09877-7

Zhu Jun, Qiu Jianxian. WENO schemes and their application as limiters for RKDG methods based on trigonometric approximation spaces[J]. Journal of Scientific Computing, 2013, 55(3): 606−644. doi: 10.1007/s10915-012-9649-9

Radakovič. The theory of approximation[J]. Monatshefte für Mathematik und Physik, 1932, 39(1): A15−A16. doi: 10.1007/BF01699111

Van Leer B. Towards the ultimate conservative difference scheme. I. The quest of monotonicity[M]. Ed. by Cabannes H, Teman R. 1973: 8.

Ha Y, Gardner C L, Gelb A, et al. Numerical simulation of high Mach number astrophysical jets with radiative cooling[J]. Journal of Scientific Computing, 2005, 24(1): 29−44. doi: 10.1007/s10915-004-4786-4

Christofi S N. The study of building blocks for essentially non-oscillatory (ENO) schemes[D]. Rhode Island: Brown University, 1996.

Artebrant R, Schroll H J. Conservative logarithmic reconstructions and finite volume methods[J]. Journal of Scientific Computing, 2005, 27(1): 294−314. doi: 10.1137/03060240X

Artebrant R, Schroll H J. Limiter-free third order logarithmic reconstruction[J]. Journal of Scientific Computing, 2006, 28(1): 359−381. doi: 10.1137/040620187

Artebrant R. Third order accurate non-polynomial reconstruction on rectangular and triangular meshes[J]. Journal of Scientific Computing, 2007, 30(2): 193−221. doi: 10.1007/s10915-005-9063-7

Liang Tian, Fu Lin. A new type of non-polynomial based TENO scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2024, 497: 112618. doi: 10.1016/j.jcp.2023.112618

Guo Jingyang, Jung J H. A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method[J]. Applied Numerical Mathematics, 2017, 112: 27−50. doi: 10.1016/j.apnum.2016.10.003

Aboiyar T, Georgoulis E H, Iske A. Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction[J]. SIAM Journal on Scientific Computing, 2010, 32(6): 3251−3277. doi: 10.1137/100792573

Bigoni C, Hesthaven J S. Adaptive WENO methods based on radial basis function reconstruction[J]. Journal of Scientific Computing, 2017, 72(3): 986−1020. doi: 10.1007/s10915-017-0383-1

Jeong B, Yang H, Yoon J. Development of a WENO scheme based on radial basis function with an improved convergence order[J]. Journal of Computational Physics, 2022, 468: 111502. doi: 10.1016/j.jcp.2022.111502

Deng Xi, Shimizu Y, Xiao Feng. A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm[J]. Journal of Computational Physics, 2019, 386: 323−349. doi: 10.1016/j.jcp.2019.02.024

Sun Ziyao, Inaba S, Xiao Feng. Boundary variation diminishing (BVD) reconstruction: A new approach to improve Godunov schemes[J]. Journal of Computational Physics, 2016, 322: 309−325. doi: 10.1016/j.jcp.2016.06.051

Deng Xi, Xie Bin, Loubere R, et al. Limiter-free discontinuity capturing scheme for compressible gas dynamics with reactive fronts[J]. Computers Fluids, 2018, 171: 1−14. doi: 10.1016/j.compfluid.2018.05.015

Deng Xi, Inaba S, Xie Bin, et al. High fidelity discontinuityresolving reconstruction for compressible multiphase flows with moving interfaces[J]. Journal of Computational Physics, 2018, 371: 945−966. doi: 10.1016/j.jcp.2018.03.036

Xiao Feng, Honma Y, Kono T. A simple algebraic interface capturing scheme using hyperbolic tangent function[J]. International Journal for Numerical Methods in Fluids, 2005, 48(9): 1023−1040. doi: 10.1002/fld.975

Marquina A. Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws[J]. SIAM Journal on Scientific Computing, 1994, 15(4): 892−915. doi: 10.1137/0915054

Thewlis J, Stevens D, Power H, et al. 4-Dimensional local radial basis function interpolation of large, uniformly spaced datasets[J]. Computer Methods and Programs in Biomedicine, 2023, 228: 107235. doi: 10.1016/j.cmpb.2022.107235

Niazi M, Rahbar K, Sheikhan M, et al. Customized RBF kernel graph-cut for weak boundary image segmentation[J]. Signal, Image and Video Processing, 2023, 17(6): 3211−3219. doi: 10.1007/s11760-023-02546-7

Wurzberger F, Schwenker F. Learning in deep radial basis function networks[J]. Entropy, 2024, 26(5). doi: 10.3390/e26050368

Hesthaven J S, Monkeberg F. Entropy stable essentially nonoscillatory methods based on RBF reconstruction[J]. Esaim Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique, 2019, 53(3): 925−958. doi: 10.1051/m2an/2019011

Deng Xi. A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order[J]. Journal of Computational Physics, 2023, 481: 112052. doi: 10.1016/j.jcp.2023.112052

Mao Meiliang, Jiang Yi, Min Yaobing, et al. A survey of geometry conservation law for high-order finite difference method[J]. Acta Aerodynamica Sinica, 2021, 39(1): 157−167(in Chinese). doi: 10.7638/kqdlxxb-2020.0111

Aràndiga F, Martí M C, Mulet P. Weights design for maximal order WENO schemes[J]. Journal of Scientific Computing, 2014, 60(3): 641−659. doi: 10.1007/s10915-013-9810-0

Hayman W K, Lingham E F. Interpolation and approximation[G]// Research Problems in Function Theory: Fiftieth Anniversary Edition. Cham: Springer International Publishing, 2019: 231−240. doi: 10.1007/978-3-030-25165-9_9

Newton I. Philosophiæ naturalis principia mathematica[M]. Editio Tertia, Aucta Emendata. Londini : Apud G. J. Innys, 1726.

Pirozzoli S. On the spectral properties of shock-capturing schemes[J]. Journal of Computational Physics, 2006, 219(2): 489−497. doi: 10.1016/j.jcp.2006.07.009

Zhang Hanxin. Implicit, non-oscillatory containing no free parameters and dissipative (INND) scheme[J]. Applied Mathematics and Mechanics, 1991, 12(1): 97−100(in Chinese).

Kemm F. A Comparative study of TVD-limiters-well-known limiters and an introduction of new ones[J]. International Journal for Numerical Methods in Fluids, 2011, 67(4): 404−440. doi: 10.1002/fld.2357

Micchelli C A. Interpolation of scattered data: Distance matrices and conditionally positive definite functions[J]. Constructive Approximation, 1986, 2(1): 11−22. doi: 10.1007/BF01893414

Shyue K M, Xiao Feng. An Eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach[J]. Journal of Computational Physics, 2014, 268: 326−354. doi: 10.1016/j.jcp.2014.03.010

Cassidy D A, Edwards J R, Tian M. An investigation of interface-sharpening schemes for multi-phase mixture flows[J]. Journal of Computational Physics, 2009, 228(16): 5628−5649. doi: 10.1016/j.jcp.2009.02.028

Xiao Feng, II S, Chen Chungang. Revisit to the THINC scheme: A simple algebraic VOF algorithm[J]. Journal of Computational Physics, 2011, 230(19): 7086−7092. doi: 10.1016/j.jcp.2011.06.012

Fu Lin, Hu X Y, Adams N A. A family of high-order targeted ENO schemes for compressible-fluid simulations[J]. Journal of Computational Physics, 2016, 305: 333−359. doi: 10.1016/j.jcp.2015.10.037

Fu Lin, Hu X Y, Adams N A. Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws[J]. Journal of Computational Physics, 2017, 349: 97−121. doi: 10.1016/j.jcp.2017.07.054

Fu Lin, Hu X Y, Adams N A. A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2018, 374: 724−751. doi: 10.1016/j.jcp.2018.07.043

Fu Lin. Very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 387. doi: 10.1016/j.cma.2021.114193

Zhang Ke, Shen Yiqing. Interface capturing schemes based on sigmoid functions[J]. Computers Fluids, 2024, 280: 106352. doi: 10.1016/j.compfluid.2024.106352

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Numerical and experimental study on aerodynamic design for the rear wing of an electric sports car

Numerical simulation on gust responses of high aspect ratio aircraft

An Ac-criteria-based optimal input design method

Research progress on noise reduction of centrifugal fans

Research on the characteristics of water droplets impacting moving superhydrophobic surfaces and its regulation

Related Author

No data

Related Institution

No data

AI问答

⁰