基于映射函数的新型五阶WENO格式

doi:10.7527/S1000-6893.2021.26155

Abstract

Abstract: Differential formats with high accuracy and high resolution are critical for numerical simulation of complex flow fields. To overcome the degradation defects of WENO-JS and WENO-Z at the first and second order extreme points of the flux function, a new mapping function (Pe) is designed and applied to the fifth order WENO scheme based on the idea of weighted coefficient reconstruction. The analyses of Approximate Dispersion Relations (ADR) indicate a smaller dispersion error and numerical dissipation of WENO-Pe than WENO-JS, WENO-Z, and other mapping function-based WENO schemes. We conduct numerical simulation in the new scheme and other schemes for 1D cases of the deformed Gaussian wave problem, Sod excitation tube problem, Lax excitation tube problem, and Shu-Osher problem, and 2D cases of the Riemann problem, Rayleigh-Taylor shock-density instability problem, and double Mach reflection problem. The results show that WENO-Pe has stronger ability to capture intermittency and higher resolution with the same order, thereby suitable for numerical simulation of complex flow fields.

Key words: WENO, high precision, high resolution, mapping function, Euler equation

CLC Number:

LIU Bo, LI Shiyao, CHEN Jiayu, CHENG Qihao, SHI Xiaotian. New fifth order WENO scheme based on mapping functions[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(12): 126155-126155.

References

[1] SENGUPTA T K, GANERWAL G, DIPANKAR A. High accuracy compact schemes and Gibbs' phenomenon[J]. Journal of Scientific Computing, 2004, 21(3):253-268.
[2] HARTEN A. High resolution schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 1983, 49(3):357-393.
[3] COCKBURN B, SHU C W. The Runge-Kutta discontinuous Galerkin method for conservation laws V[J]. Journal of Computational Physics, 1998, 141(2):199-224.
[4] HUYNH H T. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods[C]//18th AIAA Computational Fluid Dynamics Conference. Reston:AIAA, 2007.
[5] 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.
[6] SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1988, 77(2):439-471.
[7] LIU X D, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics, 1994, 115(1):200-212.
[8] JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1):202-228.
[9] 李康, 刘娜, 何志伟, 等. 一种基于双界面函数的界面捕捉方法[J]. 力学学报, 2017, 49(6):1290-1300. LI K, LIU N, HE Z W, et al. A new interface capturing method based on double interface functions[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(6):1290-1300(in Chinese).
[10] 李超群, 李易, 张晨曦, 等. 沿流向微结构沟槽流场直接数值模拟[J]. 航空学报, 2020, 41(11):123628. LI C Q, LI Y, ZHANG C X, et al. Direct numerical simulation of flow field over streamwise micro riblets[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(11):123628(in Chinese).
[11] LI H B, JIANLING L, CHA X, et al. Investigation of hot jet on active control of oblique detonation waves[J]. Chinese Journal of Aeronautics, 2020, 33(3):861-869.
[12] 刘君, 韩芳, 魏雁昕. 应用维数分裂方法推广MUSCL和WENO格式的若干问题[J]. 航空学报, 2022, 43(3):125009. LIU J, HAN F, WEI Y X. MUSCL and WENO schemes problems generated by dimension splitting approach[J]. Acta Aeronautica et Astronautica Sinica, 2022, 43(3):125009(in Chinese).
[13] 杨理, 岳连捷, 张新宇. 斜爆轰波的波角和法向速度-曲率关系初探[J]. 航空学报, 2020, 41(11):123701. YANG L, YUE L J, ZHANG X Y. Preliminary study on wave angle and normal velocity-curvature relation of oblique detonation wave[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(11):123701(in Chinese).
[14] 杜钰锋, 林俊, 王勋年, 等. 亚声速风洞可压缩流体扰动模态分析[J]. 航空学报, 2021, 42(6):124424. DU Y F, LIN J, WANG X N, et al. Analysis of modes of disturbances in compressible fluid in subsonic wind tunnel[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(6):124424(in Chinese).
[15] 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.
[16] 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.
[17] 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.
[18] YAN Z G, LIU H Y, MAO M L, et al. New nonlinear weights for improving accuracy and resolution of weighted compact nonlinear scheme[J]. Computers & Fluids, 2016, 127:226-240.
[19] BHISE A A, RAJU G N, RATHAN S, et al. An efficient hybrid WENO scheme with a problem independent discontinuity locator[J]. International Journal for Numerical Methods in Fluids, 2019, 91(1):1-28.
[20] ZHANG S H, ZHU J, SHU C W. A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes[J]. Advances in Aerodynamics, 2019(1):307-331.
[21] 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.
[22] FAN P. High order weighted essentially nonoscillatory WENO-η schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2014, 269:355-385.
[23] 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.
[24] FENG H, HUANG C, WANG R. An improved mapped weighted essentially non-oscillatory scheme[J]. Applied Mathematics and Computation, 2014, 232:453-468.
[25] FENG H, HU F X, WANG R. A new mapped weighted essentially non-oscillatory scheme[J]. Journal of Scientific Computing, 2012, 51(2):449-473.
[26] LI Q, LIU P X, ZHANG H X. Piecewise polynomial mapping method and corresponding WENO scheme with improved resolution[J]. Communications in Computational Physics, 2015, 18(5):1417-1444.
[27] 刘朋欣, 李沁, 张涵信. 基于映射函数的中心型三阶格式[J]. 空气动力学学报, 2017, 35(1):71-77. LIU P X, LI Q, ZHANG H X. A kind of third order central scheme based on mapping functions[J]. Acta Aerodynamica Sinica, 2017, 35(1):71-77(in Chinese).
[28] WANG R, FENG H, HUANG C. A new mapped weighted essentially non-oscillatory method using rational mapping function[J]. Journal of Scientific Computing, 2016, 67(2):540-580.
[29] VEVEK U S, ZANG B, NEW T H. Adaptive mapping for high order WENO methods[J]. Journal of Computational Physics, 2019, 381:162-188.
[30] 张德良. 计算流体力学教程[M]. 北京:高等教育出版社, 2010:163-168. ZHANG D L. A course in computational fluid dynamics[M]. Beijing:Higher Education Press, 2010:163-168. (in Chinese).
[31] SHU C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws:NASA C-R-97-206253[J]. Washington, D.C.:NASA,1998.
[32] PIROZZOLI S. On the spectral properties of shock-capturing schemes[J]. Journal of Computational Physics, 2006, 219(2):489-497.
[33] 涂国华, 邓小刚, 毛枚良. 5阶非线性WCNS和WENO差分格式频谱特性比较[J]. 空气动力学学报, 2012, 30(6):709-712. TU G H, DENG X G, MAO M L. Spectral property comparison of fifth-order nonlinear WCNS and WENO difference schemes[J]. Acta Aerodynamica Sinica, 2012, 30(6):709-712(in Chinese).
[34] 骆信, 吴颂平. 改进的五阶WENO-Z+格式[J]. 力学学报, 2019, 51(6):1927-1939. LUO X, WU S P. An improved fifth-order WENO-Z+ scheme[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6):1927-1939(in Chinese).
[35] 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.
[36] 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.
[37] SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1988, 77(2):439-471.
[38] YEE H C, WARMING R F, HARTEN A. Implicit total variation diminishing (TVD) schemes for steady-state calculations[J]. Journal of Computational Physics, 1985, 57(3):327-360.
[39] LAX P D, LIU X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. SIAM Journal on Scientific Computing, 1998, 19(2):319-340.
[40] 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.
[41] 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.

New fifth order WENO scheme based on mapping functions

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	LI Yufei, TIAN Wei, LI Bo, ZHANG Nan. Accuracy control method and experiment of robot milling system [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(5): 625815-625815.
[2]	LIU Jun, HAN Fang, WEI Yanxin. MUSCL and WENO schemes problems generated by dimension splitting approach [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(3): 125009-125009.
[3]	LIU Jun, HAN Fang, WEI Yanxin. Numerical errors of high-order WENO schemes under specific conditions [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(2): 124940-124940.
[4]	HE Kang, LI Xinliang, LIU Hongwei. An extension of hybrid kinetic WENO method [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(1): 625909-625909.
[5]	XU Ke, GAO Zhenxun, JIANG Chongwen, LEE Chunhian. Theoretical analysis of reconstruction accuracy of characteristic-wise WENO scheme within boundary layer [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2021, 42(S1): 726362-726362.
[6]	LI Chaoqun, LI Yi, ZHANG Chenxi, TANG Shuo. Direct numerical simulation of flow field over streamwise micro riblets [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020, 41(11): 123628-123628.
[7]	LIU Jingyu, CUI Yufu, XU Meng, HE Hongyan, YIN Huan, WANG Yu. Image restoration method for TDICCD camera with micro vibration [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2018, 39(S1): 722230-722230.
[8]	YE Zhou, XU Guohua, SHI Yongjie. High-resolution numerical research on formation and evolution mechanism of rotor blade tip vortex [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(7): 520846-520846.
[9]	TIAN Yugang, YANG Gui, WU Wei. A strict geometric calibration method for airborne hyperspectral sensors aided by high resolution images [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2015, 36(4): 1250-1258.
[10]	SHEN Zhibo, ZHAO Guoqing, DONG Chunxi, HUANG Long. United Frequency and DOA Estimation Algorithm Based on Compressed Sensing [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2014, 35(5): 1357-1364.
[11]	WANG Xin, WANG Ling, ZHU Daiyin. A Novel Frequency Domain Back-projection Algorithm for Ultra-high Resolution SAR Imaging [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2014, 35(4): 1053-1063.
[12]	MAO Xinhua, ZHU Daiyin, ZHU Zhaoda. 2-D Autofocus Algorithm for Ultra-high Resolution Airborne Spotlight SAR Imaging [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, 33(7): 1289-1295.
[13]	LI Xile, YANG Yong. Numerical Simulation of the Free Rolling Motion of a Delta Wing Configuration with Aileron Deflection [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, (3): 453-462.
[14]	LIU Yin, WU Shunjun, WU Mingyu, LI Chunmao, ZHANG Huaigen. Wideband DOA Estimation Based on Spatial Sparseness [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, 33(11): 2028-2038.
[15]	GU Ning, LU Zhiliang, ZHANG Jiaqi, GUO Tongqing. CFD-based Analysis for Gust Response of Aircraft Wing [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2011, 32(5): 785-791.