Research on hybrid method based on CPR and high-resolution second order scheme

GUO Jia; ZHU Huajun; SHI Guoquan; YAN Zhenguo; SONG Songhe; TANG Lingyan

doi:10.7638/kqdlxxb-2020.0130

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Research on hybrid method based on CPR and high-resolution second order scheme

更新时间：2026-05-28

- Research on hybrid method based on CPR and high-resolution second order scheme
- Vol. 40, Issue 5, Pages: 122-132(2022)
- 作者机构：
  
  1. 中国空气动力研究与发展中心空气动力学国家重点实验室,绵阳,621000
  2. 国防科技大学理学院数学系,长沙,410073
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2020.0130
  CLC： O241.82
- Published：2022
- 稿件说明：
移动端阅览
郭嘉, 朱华君, 石国权, 等. CPR方法与高分辨率二阶格式的混合算法[J]. 空气动力学学报, 2022, 40(5): 122−132.

GUO Jia, ZHU Huajun, SHI Guoquan, et al. Research on hybrid method based on CPR and high-resolution second order scheme[J]. 2022, 40(5): 122-132.
郭嘉, 朱华君, 石国权, 等. CPR方法与高分辨率二阶格式的混合算法[J]. 空气动力学学报, 2022, 40(5): 122−132. DOI： 10.7638/kqdlxxb-2020.0130

GUO Jia, ZHU Huajun, SHI Guoquan, et al. Research on hybrid method based on CPR and high-resolution second order scheme[J]. 2022, 40(5): 122-132. DOI： 10.7638/kqdlxxb-2020.0130.

摘要

重构修正方法（correction procedure via reconstruction，CPR）具有紧致高效的优点，但对较强激波的捕捉能力还相对较弱，而加权紧致非线性格式（weighted compact nonlinear scheme，WCNS）具有很强的激波捕捉能力。将基于高阶WCNS插值的二阶格式引入到高阶CPR方法中，构造了一种高效高分辨率的混合激波捕捉格式。首先，基于非线性权偏离线性权的程度的激波侦测器侦测出问题单元，并在问题单元附近引入缓冲单元，其余单元则标记为光滑单元。然后，针对问题单元和缓冲单元采用二阶格式计算，光滑单元采用CPR方法计算，构造混合格式。通过对等熵涡问题、含激波的问题以及激波旋涡干扰问题的数值模拟，测试了混合格式的精度、激波捕捉能力和计算效率。数值模拟结果充分说明了该混合格式具有很强的激波捕捉能力，同时在光滑区具有高分辨特性，可以应用于高超声速流动问题的高效数值模拟中。相比于基于高阶WCNS插值的二阶格式，此格式具有更高的计算效率和更高的分辨率。

Abstract

Correction Procedure via Reconstruction (CPR) is a compact and efficient numerical method

but it still has drawbacks in capturing strong shocks. However

Weighted Compact Nonlinear Scheme (WCNS) can capture shocks well. By introducing second-order scheme based on high-order interpolation of WCNS into high-order CPR method

an efficient hybrid shock-capturing method with high resolution property is proposed. Firstly

shock detection indicator based on the deviation of nonlinear weights from linear weights is used to judge troubled cells. Buffer cells are introduced in the neighbors of troubled cells

and other cells are marked as smooth cells. Therefore we use the second-order scheme to calculate in troubled cells and buffer cells

and CPR to calculate in smooth cells

which is the main idea of the proposed hybrid method. By calculating several numerical problems including isentropic vortex

shock and shock-vortex interaction

the accuracy

shock capturing ability and efficiency of hybrid method are validated with reference results. Numerical results show that the proposed hybrid method has satisfying shock-capturing ability

and has high-resolution property in smooth areas

which can be efficiently applied to numerical simulations of supersonic flows. Moreover

the method is more efficient compared with the second-order scheme based on high-order interpolation of WCNS.

关键词

Keywords

references

HUYNH H T, WANG Z J, VINCENT P E. High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids[J]. Computers Fluids, 2014, 98: 209-220. DOI:10.1016/j.compfluid.2013.12.007

WANG Z J, LI Y, JIA F, et al. Towards industrial large eddy simulation using the FR/CPR method[J]. Computers Fluids, 2017, 156: 579-589. DOI:10.1016/j.compfluid.2017.04026

DENG X G, MAO M L, TU G H, et al. High-order and high accurate CFD methods and their applications for complex grid problems[J]. Communications in Computational Physics, 2012, 11(4): 1081-1102. DOI:10.4208/cicp.100510.150511s

HUYNH H T. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods[C]//18th AIAA Computational Fluid Dynamics Conference, Miami, Florida. Reston, Virginia: AIAA, 2007. doi: 10.2514/6.2007-4079

WANG Z J, GAO H Y. A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids[J]. Journal of Computational Physics, 2009, 228(21): 8161-8186. DOI:10.1016/j.jcp.2009.07.036

DU J, SHU C W, ZHANG M P. A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework[J]. Applied Numerical Mathematics, 2015, 95: 173-198. DOI:10.1016/j.apnum.2014.01.006

LI W A, WANG Q, REN Y X. A p-weighted limiter for the discontinuous Galerkin method on one-dimensional and two-dimensional triangular grids[J]. Journal of Computational Physics, 2020, 407: 109246. DOI:10.1016/j.jcp.2020.109246

BARTER G E, DARMOFAL D L. Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation[J]. Journal of Computational Physics, 2010, 229(5): 1810-1827. DOI:10.1016/j.jcp.2009.11.010

PERSSON P O, PERAIRE J. Sub-cell shock capturing for discontinuous Galerkin methods[C]//44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virginia: AIAA, 2006 doi: 10.2514/6.2006-112

CHENG J, LU Y W, LIU T G. Multi-domain hybrid RKDG and WENO methods for hyperbolic conservation laws[J]. SIAM Journal on Scientific Computing, 2013, 35(2): A1049-A1072. DOI:10.1137/110855156

ZHU H J, YAN Z G, LIU H Y, et al. High-order hybrid WCNS-CPR schemes on hybrid meshes with curved edges for conservation laws I: spatial accuracy and geometric conservation laws[J]. Communications in Computational Physics, 2018, 23(5): 1355-1392. DOI:10.4208/cicp.oa-2017-0032

GUO J, ZHU H J, YAN Z G, et al. High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws[J]. International Journal of Aerospace Engineering, 2020, Article ID 8825445. doi: 10.1155/2020/8825445

DENG X G, ZHANG H X. Developing high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2000, 165(1): 22-44. DOI:10.1006/jcph.2000.6594

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)

DENG X G. High-order finite difference schemes based on symmetric conservative metric method: decomposition, geometric meaning and connection with finite volume schemes[J]. Advances in Applied Mathematics and Mechanics, 2020, 12(2): 436-479. DOI:10.4208/aamm.oa-2017-0243

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

YAN Z G, LIU H Y, MAO M L, et al. Development of 3rd-order HWCNS and its application in hypersonic flow[J]. Acta Aeronautica et Astronautica Sinica, 2015, 36(5): 1460-1470. (in Chinese)

WU M, MARTIN M P. Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp[J]. AIAA Journal, 2007, 45(4): 879-889. DOI:10.2514/1.27021

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

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

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

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

TANG L Y, SONG S H, ZHANG H. High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws[J]. Applied Mathematics and Mechanics, 2020, 41(1): 173-192. DOI:10.1007/s10483-020-2554-8

ZHANG X X, SHU C W. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes[J]. Journal of Computational Physics, 2010, 229(23): 8918-8934. DOI:10.1016/j.jcp.2010.08.016

RAULT A, CHIAVASSA G, DONAT R. Shock-vortex interactions at high Mach numbers[J]. Journal of Scientific Computing, 2003, 19(1-3): 347-371. DOI:10.1023/A: 1025316311633

LU Q L, LIU G M, MING P J, et al. A parameter-free gradient-based limiter for the FR/CPR method on mixed unstructured meshes[C]//Proc of the AIAA Aviation 2019 Forum, Dallas, Texas. Reston, Virginia: AIAA, 2019

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