An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator

WANG Yahui

doi:10.21656/1000-0887.420194

Volume 43 Issue 7

Jul. 2022

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WANG Yahui. An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator[J]. Applied Mathematics and Mechanics, 2022, 43(7): 802-815. doi: 10.21656/1000-0887.420194

Citation:

WANG Yahui. An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator[J]. Applied Mathematics and Mechanics, 2022, 43(7): 802-815. doi: 10.21656/1000-0887.420194

Citation:

WANG Yahui. An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator[J]. Applied Mathematics and Mechanics, 2022, 43(7): 802-815. doi: 10.21656/1000-0887.420194

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An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator

doi: 10.21656/1000-0887.420194

WANG Yahui

School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, P.R.China

Received Date: 2021-07-12
Rev Recd Date: 2021-08-22
Publish Date: 2022-07-15

Abstract

Abstract

In order to meet the requirement of high accuracy and high resolution in computational fluid dynamics (CFD), a new reference smoothness indicator was proposed to reduce the numerical dissipation of the classical 3rd-order weighted essentially non-oscillatory (WENO) scheme. The construction method is different from the classical WENO-Z scheme. It is obtained through the L²-norm approximation of the derivatives of the reconstruction polynomials of the whole global stencil, and the linear combination of the derivatives of the reconstruction polynomials on the candidate sub-stencils. With this calculation method, higher-order reference smoothness indicators can be obtained than the WENO-Z scheme. In addition, different reference smoothness indicators can be obtained by change of the value of free parameter $ \varphi$. A series of numerical examples prove the effectiveness of the reference smoothness indicator.
- hyperbolic conservation law,
- WENO,
- reconstruction polynomial,
- L²-norm,
- nonlinear weight

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References(23)

References

[1]	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
[2]	HARTEN A, OSHER S. Uniformly high-order accurate non-oscillatory schemes Ⅰ[J]. SIAM Journal on Numerical Analysis, 1987, 24(2): 279-309. doi: 10.1137/0724022
[3]	HARTEN A, ENGQUIST B, OSHER S, et al. Uniformly high-order accurate essentially non-oscillatory schemes Ⅲ[J]. Journal of Computational Physics, 1987, 71: 231-303. doi: 10.1016/0021-9991(87)90031-3
[4]	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
[5]	SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes Ⅱ[J]. Journal of Computational Physics, 1989, 83(1): 32-78. doi: 10.1016/0021-9991(89)90222-2
[6]	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
[7]	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
[8]	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: 3191-3211. doi: 10.1016/j.jcp.2007.11.038
[9]	GEROLYMOS R A, SÉNÉCHAL S, 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
[10]	WANG Y H, DU Y L, ZHAO K L, et al. Modified stencil approximations for fifth-order weighted essentially non-oscillatory schemes[J]. Journal of Scientific Computing, 2019, 81(6): 898-922.
[11]	FU L, 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
[12]	WU X S, ZHAO Y X. A high-resolution hybrid scheme for hyperbolic conservation laws[J]. International Journal for Numerical Methods in Fluids, 2015, 78(3): 162-187. doi: 10.1002/fld.4014
[13]	WU X S, LIANG J H, ZHAO Y X. A new smoothness indicator for third-order WENO scheme[J]. International Journal for Numerical Methods in Fluids, 2016, 81(7): 451-459. doi: 10.1002/fld.4194
[14]	XU W Z, WU W G. An improved third-order WENO-Z scheme[J]. Journal of Scientific Computing, 2018, 75: 1808-1841. doi: 10.1007/s10915-017-0587-4
[15]	WANG Y H, DU Y L, ZHAO K L, et al. A low-dissipation third-order weighted essentially nonoscillatory scheme with a new reference smoothness indicator[J]. International Journal for Numerical Methods in Fluid, 2020, 92(9): 1212-1234.
[16]	王亚辉. 求解双曲守恒律方程的三阶修正模板WENO格式[J]. 应用数学和力学, 2022, 43(2): 224-236. (WANG Yahui. A 3rd-order modified stencil WENO scheme for solution of hyperbolic conservation law equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 224-236.(in Chinese) WANG Yahui. A 3rd-order modified stencil WENO scheme for solution of hyperbolic conservation law equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 224-236. (in Chinese))
[17]	徐维铮, 孔祥韶, 吴卫国. 基于映射函数的三阶 WENO 改进格式及其应用[J]. 应用数学和力学, 2017, 38(10): 1120-1135. (XU Weizheng, KONG Xiangshao, WU Weiguo. An improved 3rd-order WENO scheme based on mapping functions and its application[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1120-1135.(in Chinese) XU Weizheng, KONG Xiangshao, WU Weiguo. An improved 3rd-order WENO scheme based on mapping functions and its application[J]. Applied Mathematics and Mechanics, 2017, 38(10): 1120-1135. (in Chinese)
[18]	徐维铮, 吴卫国. 三阶WENO-Z格式精度分析及其改进格式[J]. 应用数学和力学, 2018, 39(8): 946-960. (XU Weizheng, WU Weiguo. Precision analysis of the 3rd-order WENO-Z scheme and its improved scheme[J]. Applied Mathematics and Mechanics, 2018, 39(8): 946-960.(in Chinese) XU Weizheng, WU Weiguo. Precision analysis of the 3rd-order WENO-Z scheme and its improved scheme[J]. Applied Mathematics and Mechanics, 2018, 39(8): 946-960. (in Chinese))
[19]	LAX P D. Weak solutions of nonlinear hyperbolic equations and their numerical computation[J]. Communications on Pure and Applied Mathematics, 1954, 7(1): 159-193. doi: 10.1002/cpa.3160070112
[20]	SOD G A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J]. Journal of Computational Physics, 1978, 27(1): 1-31. doi: 10.1016/0021-9991(78)90023-2
[21]	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
[22]	SCHULZ-RINNE C W, COLLINS J P, GLAZ H M. Numerical solution of the Riemann problem for two-dimensional gas dynamics[J]. SIAM Journal on Scientific Computing, 1993, 14(6): 1394-1414. doi: 10.1137/0914082
[23]	WOODWAED P, COLELLA P. The numerical simulation of two-dimensional fluid flow with strong shocks[J]. Journal of Computational Physics, 1984, 54(1): 447-465.