A new type of smoothness indicator for weighted essentially non-oscillatory schemes and its application

LIU Xuliang; WU Conghai; LI Hu; FAN Zhaolin

doi:10.7638/kqdlxxb-2022.0013

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A new type of smoothness indicator for weighted essentially non-oscillatory schemes and its application

更新时间：2026-05-28

- A new type of smoothness indicator for weighted essentially non-oscillatory schemes and its application
- Vol. 41, Issue 5, Pages: 20-34(2023)
- 作者机构：
  
  1. 空气动力学国家重点实验室,绵阳,621000
  2. 中国空气动力研究与发展中心,绵阳,621000
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2022.0013
  CLC： V211.3
- Published：2023
- 稿件说明：
移动端阅览
刘旭亮, 武从海, 李虎, 等. WENO格式的一种新型光滑因子及其应用[J]. 空气动力学学报, 2023, 41(5): 20−34.

LIU Xuliang, WU Conghai, LI Hu, et al. A new type of smoothness indicator for weighted essentially non-oscillatory schemes and its application[J]. 2023, 41(5): 20-34.
刘旭亮, 武从海, 李虎, 等. WENO格式的一种新型光滑因子及其应用[J]. 空气动力学学报, 2023, 41(5): 20−34. DOI： 10.7638/kqdlxxb-2022.0013

LIU Xuliang, WU Conghai, LI Hu, et al. A new type of smoothness indicator for weighted essentially non-oscillatory schemes and its application[J]. 2023, 41(5): 20-34. DOI： 10.7638/kqdlxxb-2022.0013.

摘要

光滑因子是构造WENO格式的关键，决定了WENO格式能否达到最优收敛精度以及在间断附近能否保持本质无振荡特性。针对广泛应用的五阶WENO格式，采用算子函数近似导数关系的方法，设计了一种新型光滑因子。在间断区域，与传统的光滑因子相比，新型光滑因子对间断的识别更准确。在光滑区域，新型光滑因子使得新型WENO格式的非线性权更接近线性权。理论分析和数值验证表明新型WENO格式即使在一阶临界点也能保持一致五阶精度。一维典型激波问题的数值结果表明，与现有WENO格式相比，新型WENO格式提高了短波的分辨率，降低了格式的耗散，同时能够准确识别间断。对于典型包含激波和剪切层的流动问题，新型WENO格式不仅能够准确地捕捉强激波，而且能够精细模拟剪切层和声波等流场结构。

Abstract

The smoothness indicator determines whether WENO (weighted essentially-non-oscillatory) schemes can achieve the optimal convergence accuracy and maintain the essentially non-oscillatory characteristics near discontinuities; consequently

it is the key to constructing WENO schemes. This paper proposes a new type of smoothness indicator for the widely used fifth-order WENO scheme by the operator function method. Compared to existing smoothness indicators

this new smoothness indicator has two advantages: firstly

it quantifies the smoothness of stencils and identifies discontinuities more accurately in discontinuous regions; secondly

the nonlinear weights are closer to the linear weights in smooth regions. The resulting WENO scheme maintains the optimal fifth-order accuracy at the first-order critical point and has lower dissipation

better shock-capturing capability

and higher resolution for short waves than those using existing smoothness indicators. It

therefore

captures strong shock waves and flow structures such as shear layers and sound waves more accurately

suggesting the great potential of this smoothness indicator in simulating complex flows.

关键词

Keywords

references

GODUNOV S K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[J]. Mathematicheskii Sbornik, 1959, 47(3): 271–306.

HARTEN A. High resolution schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 1983, 49(3): 357-393. DOI:10.1016/0021-9991(83)90136-5[LinkOut]

HARTEN A. On a class of high resolution total-variation-stable finite-difference schemes[J]. SIAM Journal on Numerical Analysis, 1984, 21(1): 1-23. DOI:10.1137/0721001[LinkOut]

OSHER S, CHAKRAVARTHY S. High Resolution Schemes and the Entropy Condition[J]. Upwind and High-Resolution Schemes, 1997: 104-133. DOI:10.1007/978-3-642-60543-7_7[LinkOut]

ZHANG H X. Non-oscillatory and non-free-parameter dissipation difference scheme[J]. Acta Aerodynamica Sinica, 1988, 6(2): 143-165. (in Chinese)[知网中文][知网英文]

ZHANG H X, HE G H, ZHANG L. Some important problems for high order accurate difference scheme solving gas dynamic equations[J]. Acta Aerodynamica Sinica, 1993(4): 347-356. (in Chinese)[万方]

HARTEN A, OSHER S, ENGQUIST B, et al. Some results on uniformly high-order accurate essentially nonoscillatory schemes[J]. Applied Numerical Mathematics, 1986, 2(3-5): 347-377. DOI:10.1016/0168-9274(86)90039-5[LinkOut]

HARTEN A, OSHER S. Uniformly high-order accurate nonoscillatory schemes. I[J]. SIAM Journal on Numerical Analysis, 1987, 24(2): 279-309. DOI:10.1137/0724022[LinkOut]

HARTEN A, ENGQUIST B, OSHER S, et al. Uniformly high order accurate essentially non-oscillatory schemes, III[J]. Journal of Computational Physics, 1987, 71(2): 231-303. DOI:10.1016/0021-9991(87)90031-3[LinkOut]

SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1988, 77(2): 439-471. DOI:10.1016/0021-9991(88)90177-5[LinkOut]

LIU X D, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics, 1994, 115(1): 200-212. DOI:10.1006/jcph.1994.1187[LinkOut]

SHU C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[J]. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, 1998(1697): 325-432. DOI:10.1007/BFb0096355[LinkOut]

JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228. DOI:10.1006/jcph.1996.0130[LinkOut]

BALSARA D S, SHU C W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy[J]. Journal of Computational Physics, 2000, 160(2): 405-452. DOI:10.1006/jcph.2000.6443[LinkOut]

SHI J, ZHANG Y T, SHU C W. Resolution of high order WENO schemes for complicated flow structures[J]. Journal of Computational Physics, 2003, 186(2): 690-696. DOI:10.1016/S0021-9991(03)00094-9[LinkOut]

GEROLYMOS G A, SÉNÉCHAL D, VALLET I. Very-high-order weno schemes[J]. Journal of Computational Physics, 2009, 228(23): 8481-8524. DOI:10.1016/j.jcp.2009.07.039[LinkOut]

HENRICK A K, ASLAM T D, POWERS J M. Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points[J]. Journal of Computational Physics, 2005, 207(2): 542-567. DOI:10.1016/j.jcp.2005.01.023[LinkOut]

BORGES R, CARMONA M, COSTA B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2008, 227(6): 3191-3211. DOI:10.1016/j.jcp.2007.11.038[LinkOut]

CASTRO M, COSTA B, DON W S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2011, 230(5): 1766-1792. DOI:10.1016/j.jcp.2010.11.028[LinkOut]

SERNA S, MARQUINA A. Power ENO methods: a fifth-order accurate Weighted Power ENO method[J]. Journal of Computational Physics, 2004, 194(2): 632-658. DOI:10.1016/j.jcp.2003.09.017[LinkOut]

HA Y, HO KIM C, JU LEE Y, et al. An improved weighted essentially non-oscillatory scheme with a new smoothness indicator[J]. Journal of Computational Physics, 2013, 232(1): 68-86. DOI:10.1016/j.jcp.2012.06.016[LinkOut]

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[LinkOut]

RATHAN S, G N R. An improved non-linear weights for seventh-order weighted essentially non-oscillatory scheme[J]. Computers Fluids, 2017, 156: 496-514. DOI:10.1016/j.compfluid.2017.08.023[LinkOut]

RATHAN S, RAJU G N. A modified fifth-order WENO scheme for hyperbolic conservation laws[J]. Computers Mathematics With Applications, 2018, 75(5): 1531-1549. DOI:10.1016/j.camwa.2017.11.020[LinkOut]

BAEZA A, BÜRGER R, MULET P, et al. On the efficient computation of smoothness indicators for a class of WENO reconstructions[J]. Journal of Scientific Computing, 2019, 80(2): 1240-1263. DOI:10.1007/s10915-019-00974-7[LinkOut]

WU C H, WU L, LI H, et al. Very high order WENO schemes using efficient smoothness indicators[J]. Journal of Computational Physics, 2021, 432: 110158. DOI:10.1016/j.jcp.2021.110158[LinkOut]

WU C H, WU L, ZHANG S H. A smoothness indicator constant for sine functions[J]. Journal of Computational Physics, 2020, 419: 109661. DOI:10.1016/j.jcp.2020.109661[LinkOut]

LIU X L, ZHANG S H, ZHANG H X. Development of high-order weighted compact schemes with various difference methods[J]. Computers Fluids, 2016, 136: 114-131. DOI:10.1016/j.compfluid.2016.06.003[LinkOut]

TITAREV V A, TORO E F. Finite-volume WENO schemes for three-dimensional conservation laws[J]. Journal of Computational Physics, 2004, 201(1): 238-260. DOI:10.1016/j.jcp.2004.05.015[LinkOut]

WOODWARD P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics, 1984, 54(1): 115-173. DOI:10.1016/0021-9991(84)90142-6[LinkOut]

YEE H C, SANDHAM N D, DJOMEHRI M J. Low-dissipative high-order shock-capturing methods using characteristic-based filters[J]. Journal of Computational Physics, 1999, 150(1): 199-238. DOI:10.1006/jcph.1998.6177[LinkOut]

LIU X L, ZHANG S H. Direct numerical simulation of the interaction of 2d shock wave and shear layer[J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(1): 61-75. (in Chinese)[知网中文][知网英文

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