基于映射函数的新型五阶WENO格式

刘博; 李诗尧; 陈嘉禹; 程启豪; 时晓天

doi:10.7527/S1000-6893.2021.26155

航空学报 >

2022 , Vol. 43 >Issue 12: 126155 - 126155

DOI: https://doi.org/10.7527/S1000-6893.2021.26155

流体力学与飞行力学

基于映射函数的新型五阶WENO格式

刘博 ,
李诗尧 ,
陈嘉禹 ,
程启豪 ,
时晓天

展开

1. 中国航天空气动力技术研究院, 北京 100074;
2. 中国科学院力学研究所高温气体动力学国家重点实验室, 北京 100190;
3. 中国科学院大学工程科学学院, 北京 100049;
4. 天津大学水利工程仿真与安全国家重点实验室, 天津 300072;
5. 天津大学建筑工程学院, 天津 300350;
6. 天津大学数学学院, 天津 300350

收稿日期: 2021-07-26

修回日期: 2021-08-17

网络出版日期: 2021-09-22

基金资助

国家重点研发计划（2019YFA0405300）；国家自然科学基金（11872348，11802297）

收起

New fifth order WENO scheme based on mapping functions

LIU Bo ,
LI Shiyao ,
CHEN Jiayu ,
CHENG Qihao ,
SHI Xiaotian

Expand

1. China Academy of Aerospace Aerodynamics, Beijing 100074, China;
2. State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China;
3. School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China;
4. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China;
5. School of Civil Engineering, Tianjin University, Tianjin 300350, China;
6. School of Mathematics, Tianjin University, Tianjin 300350, China

Received date: 2021-07-26

Revised date: 2021-08-17

Online published: 2021-09-22

Supported by

National Key Research and Development Program of China(2019YFA0405300); National Natural Science Foundation of China (11872348, 11802297)

Fold

摘要

研究高精度和高分辨率的差分格式对于复杂流场的数值模拟有重要意义。为了克服WENO-JS格式和WENO-Z在通量函数的一阶和二阶极值点处降阶的缺陷，基于重构权重系数的思想，设计一族映射函数并应用到五阶WENO格式中。近似色散关系表明，WENO-Pe的色散误差和数值耗散均小于WENO-JS、WENO-Z以及其他基于映射函数的WENO格式。新格式与其他格式数值模拟变形的高斯波问题，Sod激波管、Lax激波管、激波密度干扰问题等一维算例，Riemann问题、Rayleigh-Taylor不稳定性问题、双马赫反射问题等二维算例的结果表明：在精度阶相同的情况下，WENO-Pe格式拥有更良好的捕捉间断能力，分辨率更高，适合应用于复杂流场的数值模拟。

关键词： WENO; 高精度; 高分辨率; 映射函数; 欧拉方程

本文引用格式

刘博 , 李诗尧 , 陈嘉禹 , 程启豪 , 时晓天 . 基于映射函数的新型五阶WENO格式[J]. 航空学报, 2022 , 43(12) : 126155 -126155 . DOI: 10.7527/S1000-6893.2021.26155

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

参考文献

[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.

Options

文章导航

模态框（Modal）标题

摘要

本文引用格式

Abstract

参考文献