Troubled cell detection in subcell limiting for high-order CPR method on unstructured meshes
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摘要: 重构修正（correction procedure via reconstruction, CPR）是一种计算高效、模板紧致、适用于非结构网格的高阶方法，但是无法处理流场含有较强间断的问题。针对这个问题，前期基于二阶非等距非线性加权插值的激波捕捉格式以及非结构四边形网格，发展了高阶CPR方法的一种子单元限制技术。该技术首先通过侦测因子确定出问题单元，然后将问题单元基于CPR求解点划分成非等距子单元，并在子单元中采用二阶激波捕捉格式进行计算。这种子单元限制技术保证了主单元和子单元的求解点重合，可以十分方便地处理主单元和子单元之间的数据交换。问题单元的侦测是该子单元限制技术的关键步骤，侦测出的问题单元的数量以及分布情况都将会影响数值结果。基于此，本文同时考虑了单元侦测和分维侦测两种问题单元侦测方式。通过激波管问题、Shu-Osher问题、双马赫反射问题、激波-旋涡干扰等典型算例对比分析了TVB、MDHE和JST侦测因子对数值格式分辨率的影响。研究结果表明：分维侦测能够减少问题单元的区域，进而提高分辨率。Abstract: The correction procedure via reconstruction (CPR) method is an efficient and compact method, suitable for unstructured meshes, but it cannot solve the problem with strong discontinuities in flow fields. Based on the shock-capturing scheme with second-order non-uniform nonlinear weighted interpolation, we previously developed a subcell limiting technique for the high-order CPR method on two-dimensional unstructured quadrilateral meshes. Within this technique, the troubled cells are first divided into non-uniform subcells, then a second-order shock-capturing scheme is adopted for the calculation. As the solution points are overlaid between the main cells and the subcells, it is very convenient to handle the data exchange in the two layers. The detection of troubled cells is a crucial step in the subcell limiting technique, and the number and distribution of troubled cells can both affect the numerical results. Based on this, the present study considers two methods of troubled cell detecting, i.e. detecting by cell and detecting by dimension. This paper compares and analyzes the influence of different indicators, i.e. TVB, MDH, and JST, on the resolution of numerical schemes through typical benchmark cases such as the shock tube problem, the Shu-Osher problem, the double Mach reflection problem, and the shock-vortex interaction. The results show that detecting by dimension can reduce the number of troubled cells, thereby improving the resolution, which has a certain scientific value.
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图 2 二阶非等距非线性插值模板
圆点是求解点，方块是通量点
Figure 2. Stencil for the second-order nonuniform nonlinear interpolation
图 5 Sod激波管的密度分布和问题单元分布图
Figure 5. Distributions of the density and problematic cells for the Sod shock tube problem
图 6 Shu-Osher问题的密度分布和问题单元分布
Figure 6. Distributions of the density and prolematic cellsfor the Shu-Osher problem
图 8 CPR-CNNW2在不同侦测因子下求解双马赫反射问题（从1.5到21.7的30条密度等值线）
Figure 8. Density contours with 30 levels ranging from 1.5 to 21.7 for CPR-CNNW2 with different indicators in the double Mach reflection problem
图 9 双马赫反射问题的各种侦测因子下的问题单元分布图
Figure 9. Distribution of problematic cells under different indicators for the double Mach reflection problem
图 10 双马赫反射的分维混合CPR-CNNW2的计算结果
Figure 10. Numerical results of CPR-CNNW2 with detecting by dimension for the double Mach reflection problem
图 11 不同侦测因子下CPR-CNNW2求解激波-旋涡干扰问题的计算结果
Figure 11. Numerical results of CPR-CNNW2 with different indicators for the shock-vortex interaction problem
图 12 激波-旋涡干扰问题的问题单元分布
Figure 12. Distribution of problematic cells for theshock-vortex interaction problem
图 13 激波-旋涡干扰问题的参考线位置
Figure 13. Reference lines for the shock-vortex interaction problem
图 14 不同侦测因子下单元混合CPR-CNNW2格式在两条截线上的密度分布图
Figure 14. Density distribution along the two sliced lines of CPR-CNNW2 with detecting by cell for different cell indicators
图 15 单元混合下CPR-CNNW2格式问题单元占比随时间变化图
Figure 15. Time variation of the problematic cell proportion for CPR-CNNW2 with detecting by cell
图 16 分维混合CPR-CNNW2格式在两条截线上的密度分布
Figure 16. Density distribution along two sliced lines for CPR-CNNW2 with detecting by dimension
图 17 分维混合CPR-CNNW2计算激波-旋涡干扰问题的问题单元分布
Figure 17. Distribution of problematic cells of CPR-CNNW2 with detecting by dimension for the shock-vortex interaction problem
图 18 分维混合CPR-CNNW2格式的问题单元占比随时间变化图
Figure 18. Time variation of the problematic cell proportion for CPR-CNNW2 with detecting by dimension
图 19 截线y = 0.40上三种侦测因子在不同阈值下的密度分布
Figure 19. Density distributions along the sliced line at y = 0.40 for three indicators under different thresholds
图 21 截线x =1.05上CPR-CNNW2 和WENO的密度分布对比
Figure 21. Density distributions along the sliced line at x=1.05 compared between CPR-CNNW2 and WENO
表 1 单元混合与分维混合的计算时间
Table 1. Calculation time for hybrid by cell and by dimension
混合策略 问题单元占比/% 侦测时间 计算时间/s 单元混合 16.19 160.53 2 142.20 分维混合 11.42 180.75 2 130.91 
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表 2 不同侦测因子对计算时间的影响
Table 2. Influence of different indicators on the calculation time
计算格式 问题单元
占比/%侦测
时间/s计算
时间/s侦测时间
占比/%CNNW2 100 0 2 279.13 — CPR 0 0 1 904.50 — CPR-CNNW2 TVB 0 205.38 2 124.91 9.67 MDHE 0 136.20 2 031.19 6.71 JST 0 418.36 2 342.77 17.8
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[1] WANG Z J. High-order methods for the Euler and navier-stokes equations on unstructured grids[J]. Progress in Aerospace Sciences, 2007, 43(1-3): 1-41. DOI: 10.1016/j.paerosci.2007.05.001 [2] 张来平, 贺立新, 刘伟, 等. 基于非结构/混合网格的高阶精度格式研究进展[J]. 力学进展, 2013, 43(2): 202-236. doi: 10.6052/1000-0992-12-092ZHANG L P, HE L X, LIU W, et al. Reviews of high-order methods on unstructured and hybrid grid[J]. Advances in Mechanics, 2013, 43(2): 202-236. (in Chinese) doi: 10.6052/1000-0992-12-092 [3] HUYNH H T. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods[C]//Proc of the 18th AIAA Computational Fluid Dynamics Conference, Miami, Florida. Reston, Virigina: AIAA, 2007: 1-42. doi: 10.2514/6.2007-4079 [4] 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 [5] WANG Z J, GAO H Y, HAGA T. A unifying discontinuous CPR formulation for the Navier–Stokes equations on mixed grids[C]//Computational Fluid Dynamics 2010, 2011: 59-65. doi: 10.1007/978-3-642-17884-9_5 [6] VONNEUMANN J, RICHTMYER R D. A method for the numerical calculation of hydrodynamic shocks[J]. Journal of Applied Physics, 1950, 21(3): 232-237. DOI: 10.1063/1.1699639 [7] PERSSON P O, PERAIRE J. Sub-cell shock capturing for discontinuous Galerkin methods[C]// 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virigina: AIAA, 2006: 1-13. doi: 10.2514/6.2006-112 [8] 于剑, 阎超. 基于人工粘性的间断Galerkin有限元方法[J]. 空气动力学学报, 2013, 31(3): 371-375.YU J, YAN C. Discontinuous Galerkin method based on artificial viscosity[J]. Acta Aerodynamica Sinica, 2013, 31(3): 371-375. (in Chinese) [9] ZHU J, ZHONG X H, SHU C W, et al. Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes[J]. Journal of Computational Physics, 2013, 248: 200-220. DOI: 10.1016/j.jcp.2013.04.012 [10] KIM K H, KIM C. Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: part I: spatial discretization[J]. Journal of Computational Physics, 2005, 208(2): 527-569. DOI: 10.1016/j.jcp.2005.02.021 [11] MICHALAK C, OLLIVIER-GOOCH C. Accuracy preserving limiter for the high-order accurate solution of the Euler equations[J]. Journal of Computational Physics, 2009, 228(23): 8693-8711. DOI: 10.1016/j.jcp.2009.08.021 [12] QIU J X, SHU C W. A comparison of troubled-cell indicators for runge: kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters[J]. SIAM Journal on Scientific Computing, 2005, 27(3): 995-1013. DOI: 10.1137/04061372x [13] 张帆, 张力丹, 刘君, 等. 非结构网格上的TV-van Leer混合迎风格式[J]. 空气动力学学报, 2019, 37(5): 855-863. doi: 10.7638/kqdlxxb-2017.0109ZHANG F, ZHANG L D, LIU J, et al. Hybrid TV-van leer upwind scheme on unstructured grids[J]. Acta Aerodynamica Sinica, 2019, 37(5): 855-863. (in Chinese) doi: 10.7638/kqdlxxb-2017.0109 [14] PARK J S, KIM C. Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction[J]. Journal of Computational Physics, 2016, 308: 57-80. DOI: 10.1016/j.jcp.2015.12.020 [15] N. Discacciati, J. S. Hesthaven, D. Ray. Cotrolling oscillations in high-order Discontinuous Galerkin schemes using artificial viscosity tuned by neuralnet works[J]. Journal of Computational Physics, 2020, 409: 109304. DOI: 10.1016/j.jcp.2020.109304 [16] LI Y N, WANG Z J. A convergent and accuracy preserving limiter for the FR/CPR method[C]//Proc of the 55th AIAA Aerospace Sciences Meeting, Grapevine, Texas. Reston, Virginia: AIAA, 2017: 1-16. doi: 10.2514/6.2017-0756 [17] 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 [18] HUERTA A, CASONI E, PERAIRE J. A simple shock-capturing technique for high-order discontinuous Galerkin methods[J]. International Journal for Numerical Methods in Fluids, 2012, 69(10): 1614-1632. DOI: 10.1002/fld.2654 [19] DUMBSER M, ZANOTTI O, LOUBÈRE R, et al. A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws[J]. Journal of Computational Physics, 2014, 278: 47-75. DOI: 10.1016/j.jcp.2014.08.009 [20] DUMBSER M, LOUBÈRE R. A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes[J]. Journal of Computational Physics, 2016, 319: 163-199. DOI: 10.1016/j.jcp.2016.05.002 [21] KOCHI S R S P, RAMAKRISHNA M. A compact subcell WENO limiting strategy using immediate neighbors for runge-kutta discontinuous Galerkin methods for unstructured meshes[J]. Journal of Scientific Computing, 2021, 90(1): 1-24. DOI: 10.1007/s10915-021-01725-3 [22] 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, 2020: 1-13. DOI: 10.1155/2020/8825445 [23] HENNEMANN S, RUEDA-RAMÍREZ A M, HINDENLANG F J, et al. A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations[J]. Journal of Computational Physics, 2021, 426: 109935. DOI: 10.1016/j.jcp.2020.109935 [24] SONNTAG M, MUNZ C D. Efficient parallelization of a shock capturing for discontinuous Galerkin methods using finite volume sub-cells[J]. Journal of Scientific Computing, 2017, 70(3): 1262-1289. DOI: 10.1007/s10915-016-0287-5 [25] ZHU H J, LIU H Y, YAN Z G, et al. Shock capturing schemes based on nonuniform nonlinear weighted interpolation for conservation laws and their application as subcell limiters for FR/CPR[J]. 2021: arXiv: 2107.06471 [math. NA]. [26] SHI G Q, ZHU H J, YAN Z G. A priori subcell limiting approach for the FR/CPR method on unstructured meshes[EB/OL]. 2021: arXiv: 2108.03446[math.NA]. https://arXiv.org/abs/2108.03446 [27] COCKBURN B, SHU C W. TVB runge-kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework[J]. Mathematics of Computation, 1989, 52(186): 411-435. doi: 10.2307/2008474 [28] JAMESON A., SCHMIDT W, TUTKEL E. Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes. In: Fluid Dynamics and Co-located Conferences. American Institute of Aeronautics and Astronautics, 1981.doi: 10.2514/6.1981-1259 [29] KLÖCKNER A, WARBURTON T, HESTHAVEN J S. Viscous shock capturing in a time-explicit discontinuous Galerkin method[J]. Mathematical Modelling of Natural Phenomena, 2011, 6(3): 57-83. DOI: 10.1051/mmnp/20116303 [30] 傅德薰, 马延文. 计算流体力学[M]. 北京: 高等教育出版社, 2002.FU D X, MA Y W. Computational Fluid Dynamics[M]. Beijing: Higher Education Press, 2002(in Chinese). [31] LI W, 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 [32] WANG C S. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[R]. 1997. doi: 10.1023/A: 1025316311633 [33] 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 [34] 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. doi: 10.2514/6.2019-3210 [35] 骆信, 吴颂平. 改进的五阶WENO-Z+格式[J]. 力学学报, 2019, 51(6): 1927-1939. doi: 10.6052/0459-1879-19-249LUO 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) doi: 10.6052/0459-1879-19-249 -
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