基于非结构/混合网格的高阶精度格式研究进展

张来平; 贺立新; 刘伟; 李明; 张涵信

doi:10.6052/1000-0992-12-092

基于非结构/混合网格的高阶精度格式研究进展

doi: 10.6052/1000-0992-12-092

张来平^1,2, ,,
贺立新²,
刘伟^1,2,
李明²,
张涵信²

1 空气动力学国家重点实验室, 四川绵阳 621000
2 中国空气动力研究与发展中心, 四川绵阳 621000

基金项目: 国家重点基础研究发展计划(973) (2009CB723802)和国家自然科学基金(91016001, 91016011, 91130029)资助项目

详细信息

作者简介:
张来平, 1968 年出生, 1990 年毕业于中国科学技术大学近代力学系, 1997 年中国空气动力研究与发展中心博士. 现任中国空气动力研究与发展中心计算空气动力研究所研究员, 空气动力学学会理事, 中国力学学会青年工作委员会委员. 主要从事非结构/混合网格生成技术、基于非结构/混合网格的计算格式、非定常流动机理等方面的研究与应用工作.

通讯作者:
张来平

中图分类号: V211.3
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出版历程
- 收稿日期: 2012-08-28
- 修回日期: 2013-03-19
- 刊出日期: 2013-03-25

Reviews of high-order methods on unstructured and hybrid grid

ZHANG Laiping^{1,2
, ,},
HE Lixin²,
LIU Wei^1,2,
LI Ming²,
ZHANG Hanxin²

1 State Key Laboratory of Aerodynamics, Sichuan, Mianyang 621000, China
2 China Aerodynamics Research and Development Center, Sichuan, Mianyang 621000, China

Funds: The project was supported by the National Basic Research Program of China (973 Program) (2009CB723802), the Natural Science Foundation of China (91016001, 91016011, 91130029).

More Information

Corresponding author: ZHANG Laiping

摘要

摘要: 尽管以二阶精度格式为基础的计算流体力学(CFD) 方法和软件已经在航空航天飞行器设计中发挥了重要的作用, 但是由于二阶精度格式的耗散和色散较大, 对于湍流、分离等多尺度流动现象的模拟, 现有成熟的CFD 软件仍难以给出满意的结果, 为此CFD 工作者发展了众多的高阶精度计算格式. 如果以适应的计算网格来分类, 一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法. 由于非结构/混合网格具有良好的几何适应性, 基于非结构/混合网格的高阶精度格式近年来备受关注. 本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展, 重点介绍了空间离散方法, 主要包括k-Exact 和ENO/WENO 等有限体积方法, 间断伽辽金(DG) 有限元方法, 有限谱体积(SV) 和有限谱差分(SD) 方法, 以及近来发展的各种DG/FV 混合算法和将各种方法统一在一个框架内的CPR (correctionprocedure via reconstruction) 方法等. 随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题, 包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等. 在综述过程中, 介绍了各种方法的优势与不足, 其间介绍了作者发展的基于"静动态混合重构" 的DG/FV 混合算法. 最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.
- 非结构网格 /
- 混合网格 /
- 高阶精度格式 /
- 有限体积方法 /
- 间断伽辽金方法 /
- 有限谱体积方法 /
- 有限谱差分方法 /
- DG/FV 混合算法
Abstract: While 2nd order methods are dominant in most compressible flow simulations, many types of problems, such as computational aeroacoustics (CAA), vortex-dominant flows and large eddy simulation (LES) of turbulent flows, call for higher order accuracy (third order and more). The main deficiency of widely available, second-order methods for the accurate simulations of the above-mentioned flows is the numerical diffusion and dissipation of vorticity to unacceptable level. Applications of high-order accu- rate, low-diffusion and low dissipation numerical methods can significantly alleviate this deficiency of the traditional second order methods, improve predictions of vortical and other complex, separated, unsteady flows. On the other hand, for complex configurations, the structured/unstructured hybrid grid technique presents the trend of grid generation technique. Therefore the high-order methods on unstructured and hybrid (or mixed) grids are paid much more attention in recent years. In this paper, the high-order methods on unstructured and hybrid grids are reviewed comprehensively, including the k-exact finite volume (FV) methods, the FV methods based on ENO and WENO reconstruction, the discontinuous Galerkin (DG) finite element method, the finite spectral volume (SV) methods, the finite spectral dif- ference (SD) method, DG/FV hybrid methods, the unified method based on correction procedure via reconstruction (CPR). The main ideas of these high-order methods are introduced, and the advantages and disadvantages of these methods are discussed. In addition, some important issues for simulations of complex geometry are discussed, including the treatment of curved boundary, the detector of discontinu-ity and high-order limiters, implicit iteration methods, and hp-multigrid approaches. We believe that the high-order methods on unstructured and hybrid grids will play more and more important role on more accurate simulations of realistic airspace vehicles and study of complex flow mechanism in the future.
- unstructured grid /
- hybrid grid /
- high-order method /
- finite element method /
- discontinuous Galerkin method /
- finite spectral volume method /
- finite spectral difference method /
- hybrid DG/FV methods

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参考文献(177)

[1]	Abgrall R. 1994.On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys., 114: 45-58
[2]	Aftosmis M, Gaitonde D, Tavares T S. 1995. Behavior of linear reconstruction techniques on unstructured meshes. AIAA J., 33 (11): 2038-2049
[3]	Anderson W K. 1994. A grid generation and flow solution method for the Euler equations on unstructured grids. J. Comput. Phys., 110: 23-38
[4]	Argyris J, St. Doltsinis I, Friz H. 1989. Hermes shuttle: Exploration of reentry aerodynamics. Computer Meth. Appli. Mech. Eng., 73: 1-51
[5]	Argyris J, St. Doltsinis I, Friz H. 1990. Studies on computational reentry aerodynamics. Computer Meth. Appli. Mech. Eng., 81: 257-289
[6]	Arminjon P. 1993. Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys., 106: 176-198
[7]	Baker T J. 2005. Mesh generation: Art or science? Prog. Aerospace Sci., 41: 29-63
[8]	Balan A, May G, Schoberl J. 2012. A stable high-order spectral difference method for hyperbolic conservation laws on triangular elements, J. Comput. Phys., 231: 2359-2375
[9]	Barth T J, Jespersen D C. 1989. The design and application of upwind schemes on unstructured meshes. AIAA Paper 89-0366
[10]	Barth T J, Frederickson P O. 1990. High-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper 90-0013
[11]	Bassi F, Rebay S. 1994. Accurate 2D computations by means of a high order discontinuous finite element method. In XIV International Conference on Numerical Methods in Fluid Dynamics; Lecture Notes in Physics, 453: 234-240, Springer, Berlin.
[12]	Bassi F, Rebay S. 1997a. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131 (1): 267-279
[13]	Bassi F, Rebay S. 1997b. High order accurate discontinuous finite element solution of 2D Euler equations. J. Comput. Phys., 138: 251-285
[14]	Bassi F, Rebay S. 2002. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. Int. J. Numer. Methods Fluids, 40 (1): 197-207
[15]	Bassi F, Crivellini A, Rebay S, et al. 2005. Discontinuous Galerkin solutions of the Reynolds-averaged Navier-Stokes and k-w turbulence model equations. Computers & Fluids, 34 (4-5): 507-540
[16]	Biswas R, Devine K D, Flaherty J E. 1994. Parallel, adaptive finite element methods for conservation laws. Appli. Numer. Math., 14: 255-283
[17]	NASA Langley. 2012. CFD resources at NASA Langley: Computational Fluid Dynamics, http://aaac.larc.nasa.gov/tsab/cfdlarc/
[18]	陈坚强. 1995. 超声速燃烧流场及旋涡流动的数值模拟. [博士论文]. 中国空气动力研究与发展中心. (Chen J Q. 1995. The numerical simulation of supersonic combustion flow field and vortex flow. [PhD Dissertation]. CARDC (in Chinese))
[19]	Chen Q Y. 2006a. Partitions of a simplex leading to accurate spectral (finite) volume reconstruction. SIAM J. Sci. Comput., 27 (4): 1458-1470
[20]	Chen Q Y. 2006b. Partitions for spectral finite volume reconstruction in the tetrahedron. SIAM J. Sci. Comput., 29 (3): 299-319
[21]	Cockburn B, Lin S Y, Shu C W. 1989. TVD Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J. Comput. Phys., 84:90-113
[22]	Cockburn B, Shu C W. 1989. TVD Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math. Comp., 52: 411-435
[23]	Cockburn B, Hou S, Shu C W. 1990. TVD Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp., 54: 545-581
[24]	Cockburn B, Shu C W. 1994. Nonlinearly stable compact schemes for shock calculations. SIAM J. Numer. Anal., 31 (3): 607-630
[25]	Cockburn B, Shu C W. 1998. The local discontinuous Galerkin method for time-dependent convection diffusion system. SIAM J. Numer. Anal., 35: 2440-2463%
[26]	Cockburn B, Karniadakis G E, Shu C W. 1999. Discontinuous Galerkin methods. Springer, Berlin.
[27]	Collis S S. 2002. Discontinuous Galerkin methods for turbulence simulation. Center for Turbulence Research Proceedings of the Summer Program, pp. 115-167
[28]	Cook A W, Cabot W H. 2004. A high-wave number viscosity for high-resolution numerical method. J. Comput. Phys., 195 (2): 594-601
[29]	Courant R, Friedrichs K O, Lewy H. 1967. On the partial difference equations of mathematical physics. IBM J., March, 11 (2): 215-234
[30]	蔡庆东, 温功碧. 1994. 二维可压无粘流的自适应流量修正有限元解, 航空学报, 15 (11): 1291-1297. (Cai Q D, Wen G B. 1994. Finite element flux corrected transport (FEM-FCT) solution of two-dimensional Euler equation on adaptive triangular mesh. Acta Aeronautica & Astronautica Sinica, 15 (11): 1291-1297. (in Chinese))
[31]	蔡庆东. 1996. 新型NND有限元方法和三维FCT有限元技术的研究. [博士论文]. 北京: 北京大学 (Cai Q D. 1996. The study of a new NND finite element scheme and three dimensional FCT finite element technology. [PhD Dissertation]. Beijing: Peking University. (in Chinese))
[32]	Deng X G, Maekawa H. 1997. Compact high-order accurate nonlinear schemes. J. Comput. Phys., 130: 77-91
[33]	Deng X G, Zhang H X. 2000. Developing high-order accurate nonlinear schemes. J. Comput. Phys., 165 (1): 22-44
[34]	Deng X G. 2001. High-order accurate dissipative weighted compact nonlinear schemes. Science in China, Series A, 31 (12): 1104-1117
[35]	Deng X G, Mao M L, Tu G H, et al. 2010. Extending weighted compact nonlinear schemes to complex grids with characteristic-based interface conditions. AIAA J., 48 (12): 2840-2851
[36]	Deng X G, Mao M L, Tu G H, et al. 2011. Geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys., 230: 1100-1115
[37]	Deng X G, Min Y B, Mao M L, et al. 2013. Further studies on geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys., 239: 90-111
[38]	Desideri J A, Dervieux A. 1988. Compressible flow solvers using unstructured grids. VKI Lecture Series, 1988-05, VAN Karman Inst.; for Fluid Dynamics, Rhode St. Genese, Belgium pp. 1-115
[39]	Donea J. 1984. A Taylor-Galerkin method for convective transport problems. Int. J. Numer. Methods Eng., 20:101-120
[40]	Donea J, Quartapelle L, Selmin V. 1987. An analysis of time discretization in the finite element solution of hyperbolic problems. J. Comput. Phys.,70: 463-499
[41]	段占元, 童秉纲, 姜贵庆. 1997. 有限差分－有限元混合方法及其在气动热计算中的应用. 空气动力学学报, 15 (4): 5-10 (Duan Z Y, Tong B G, Jiang G Q. 1997. A high resolution hybrid finite difference-finite element method and its applications to aerodynamic heating calculation. Acta Aerodynamica Sinica, 15 (4): 5-10. (in Chinese))
[42]	Dumbser M, Balsara D S, Toro E F. 2008. A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys., 227: 8209-8253
[43]	Dumbser M, Zanotti O. 2009. Very high order P_NP_M schemes on unstructured meshes for the resistive relativistic MHD equations. J. Comput. Phys., 228: 6991-7006
[44]	Dumbser M. 2010. Arbitrary high order P_NP_M schemes on unstructured meshes for the compressible Navier-Stokes equations. Computers and Fluids, 39: 60-76%
[45]	Durlofsky L J, Enquist B, Osher S. 1992. Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys., 98: 64
[46]	Ekaterinaris J A. 2005. High-order accurate, low numerical diffusion methods for aerodynamics. Prog. Aerospace Sci., 41: 192-300
[47]	冯康. 1965. 基于变分原理的差分格式. 应用数学与计算科学, 2 (4): 237-261. (Feng K. 1965. The finite difference scheme based on variational principle. Commun. appli. Math. Comput., 2 (4): 237-261 (in Chinese))
[48]	Friedrich O. 1998. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 144: 194-212%
[49]	Frink N T. 1994. Recent progress toward a three dimensional unstructured Navier-Stokes flow solver. AIAA Paper 94-0061
[50]	Fu D X, Ma Y W. 1997. A high-order accurate difference scheme for complex flow fields. J. Comput. Phys., 134: 1-15
[51]	Gao H Y, Wang Z J, Liu Y. 2010. A study of curved boundary representations for 2D high order Euler solvers. SIAM J. Sci. Comput., 44: 323-336
[52]	Harris R, Wang Z J, Liu Y. 2007. Efficient implementation of high-order spectral volume method for multidimensional conservation laws on unstructured grids. AIAA Paper 2007-912
[53]	Harten A. 1983. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49: 357-393
[54]	Harten A., Lax P D, van Leer B. 1983. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25: 35-61
[55]	Harten A, Engquist B, Osher S, et al. 1986. Some results on uniformly high order accurate essentially non-oscillatory schemes. Appli. Numer. Math., 2: 347-377
[56]	Harten A, Engquist B, Osher S, et al. 1987. Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys., 71:231-303
[57]	Hassan O, Morgan K, Peraire J. 1990. An implicit finite element method for high speed flows. AIAA Paper 90-0402%
[58]	He L X, Zhang L P, Zhang H X. 2006. A finite element/finite volume mixed solver on hybrid grids. In: Proc. of The Fourth International Conference on Computational Fluid Dynamics, Ghent, Belgium
[59]	贺立新, 张来平, 张涵信. 2007. 间断Galerkin有限元和有限体积混合计算方法研究. 力学学报, 39 (1): 15-21 (He L X, Zhang L P, Zhang H X. 2007. A finite element/finite volume mixed solver on hybrid grids. Chinese Journal of Theoretical and Applied Mechanics, 39 (1): 15-21 (in Chinese))
[60]	贺国宏. 1994. 三阶ENN格式及其在高超声速粘性复杂流场求解中的应用. [博士论文]. 中国空气动力研究与发展中心 (He G H. 1994. The third-order ENN scheme and its application in hypersonic viscous complex flow field simulations. [PhD Dissertation]. CARDC. (in Chinese))
[61]	Heinrich J C. 1977. An upwind finite element schemes for two-dimensional convective transport equations. Int. J. Numer. Meth. .Eng., 11:131-143
[62]	Hoffmann M, Munz C D, Wang Z J. 2012. Efficient Implementation of the CPR Formulation for the Navier-Stokes Equations on GPUs. In: Proceedings of the 7th International Conference on Computational Fluid Dynamics, ICCFD7-2603.
[63]	Hu C, Shu C W. 1999. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 150: 97-127
[64]	Hughes T J R, Brooks A. 1979. A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows, ASME, New York.
[65]	Hughes T J R, Mallet M. 1986. A new finite element formulation for CFD IV: A discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng., 58 (3): 329-356%
[66]	Hughes T J R. 1987. Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluid, 7:1261-1275
[67]	Hughes T J R, Franca L P, Hulbert G M. 1989. A new finite element formulation for computational fluid dynamics VIII: The Galerkin least squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng., 73: 173-89
[68]	Huynh HT. 2007. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 2007-4079
[69]	Huynh HT. 2009. A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. AIAA Paper 2009-403
[70]	Huynh HT. 2011. High-order methods by correction procedures using reconstructions. In: Wang Z J, eds. Adaptive High-order Methods in Computational Fluid Dynamics, World Scientific, Singapore, pp. 391-422
[71]	黄兆林, 李宏伟, 毛国良. 1994. 再入飞行器高超声速绕流的一步和二步有限元计算. 空气动力学学报, 12 (2): 213-218 (Huang Z L, Li H W, Mao G L. 1994. One step and two step finite element computations for the hypersonic flow past a re-entry vehicle. Acta Aerodynamica Sinica, 12 (2): 213-218. (in Chinese))
[72]	Jaffre J, Johnson C, Szepessy A. 1995. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Meth. Appli. Sci., 5: 367-386
[73]	Jameson A, Schmidt W, Turkle E. 1981. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA Paper 81-1259
[74]	Jiang B N, Povinelli L A. 1990. Least-square finite element method for fluid dynamics. Computer Meth. Appli. Mecha. Eng., 81:13-37
[75]	Jiang G, Shu C W. 1996. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126 (1): 202-228
[76]	Kawai S, Lele S K. 2008. Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. Comput. Phys., 227: 9498-9526
[77]	Kim K, Hong L, Joon H, et al. 1997. An improvement of AUSM schemes by introducing the pressure-based weight functions. Computers & Fluids, 26 (5): 505-524
[78]	Klaij C M, Raalte M H, Ven H, et al.. 2007. h-multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 227: 1024-1045
[79]	Kopriva D A. 1998. A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations. J. Comput. Phys.,143:125-148
[80]	Krivodonova L, Xin J, Remacle J F, et al. 2004. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appli. Numer. Math., 48: 323-338%
[81]	Krivodonova L, Berger M. 2006. High-order accurate implementation of solid wall boundary condition in curved geometries. J. Comput. Phys., 211: 492-512
[82]	Krivodonova L. 2007. Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys., 226: 879-896
[83]	Lele S K. 1992. Compact finite difference scheme with spectral-like resolution. J. Comput. Phys., 103 (1): 16-42
[84]	Liou M S. 2000. Mass flux schemes and connection to shock instability. J. Comput. Phys., 160: 623-648
[85]	Li Q, Guo Q L, Sun D, et al. 2012a. On the bandwidth optimization and improvement of the nonlinear procedures of WENO-type schemes. In: Proc. of the Ninth Asian CFD Conferences, Nanjing.
[86]	Li Q, Sun D, Zhang H X. 2012. Investigations on massive separation flow and new-typed cross-flow vortices of the 76/40 double-delta wing by DES. In: Proc. of the Ninth Asian CFD Conferences, Nanjing.
[87]	李沁, 郭启龙, 孙东, 等. 2012b. 基于带宽优化和非线性技术的新四阶、五阶格式. 见: 第15届全国CFD会议文集, 2012年8月3-6, 山东烟台. (Li Q, Guo Q L, Sun D and Zhang H X. 2012b. Newly developed 4th-order and 5th-order schemes based on the bandwidth optimization and the nonlinear techniques. In: Proc. of the 15th National CFD Conference. 2012-08-3 6, Yantai, Shandong, China (in Chinese))
[88]	刘儒勋, 舒其望. 2003. 计算流体力学若干新方法. 北京: 科学出版社. (Liu R X, Shu C W. 2003. Some New Methods in Computational Fluid Dynamics. Beijing: Science Press (in Chinese))
[89]	刘伟, 张来平, 赫新, 等. 2012. 基于Newton/Gauss-Seidel迭代的高阶精度DGM隐式计算方法研究. 力学学报, 44 (4): 792-796 (Liu W, Zhang L P, He X, et al. 2012. An implicit algorithm for discontinuous Galerkin method based on Newton/Gauss-Seidel iterations. Chinese Journal of Theoretical and Applied Mechanics, 44 (4): 792-796. (in Chinese))
[90]	Liu X D, Osher S, Chan T. 1994. Weighted essentially non oscillatory schemes. J. Comput. Phys., 115: 200-212%
[91]	Liu X D, Osher S. 1998. Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids. J. Comput. Phys., 142: 304-308
[92]	Liu Y, Vinokur M, Wang Z J. 2006a. Discontinuous spectral difference method for conservation laws on unstructured grids. J. Comput. Phys., 216: 780-801%
[93]	Liu Y, Vinokur M, Wang Z. J.. 2006b. Discontinuous spectral difference method for conservation laws on unstructured grids. In: Proceedings of the 3rd International Conference on Computational Fluid Dynamics, Toronto, Canada, July 12-16.
[94]	Liu Y, Vinokur M, Wang Z J. 2006c. Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys., 212: 454-472
[95]	Li W N, Ren Y X, Lei G D, et al. 2012. The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids. J. Comput. Phys., 230: 4053-4077
[96]	Löhner R, Morgan K, Zienkiewicz O C. 1985. An adaptive finite element procedure for compressible high speed flows. Computer Methods Appli. Mecha. Eng., 51: 441-465
[97]	Luo H, Baum J D, Löhner R. 2005. High-Reynolds number viscous computations using an unstructured-grid method. J. Aircraft, 42 (2): 483-492
[98]	Luo H, Baum J D. 2006. A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys., 211: 767-783
[99]	Luo H, Baum J D, Löhner R. 2007. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids, J. Comput. Phys., 225: 686-713
[100]	Luo H, Baum J D, Löhner R. 2008. A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comput. Phys., 227: 8875-8893
[101]	Luo H, Luo L P, Nourgaliev R, et al. 2010. A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids. J. Comput. Phys., 229: 6961-6978
[102]	Luo H, Luo L P, Ali A, et al. 2011. A parallel, reconstructed discontinuous Galerkin method for the compressible flows on arbitrary grids. Commun. Comput. Phys., 9 (2): 363-389
[103]	Mavriplis D J. 1992. Three dimensional unstructured multigrid for the Euler equations. AIAA J., 30 (7): 1753-1761
[104]	Mavriplis D J. 1995. Multigrid techniques for unstructured meshes. ICASE Report, No. 95-27
[105]	Mavriplis D J. 1998. Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. J. Comput. Phys., 145: 141-165
[106]	May G, Jameson A. 2006. A spectral difference method for the Euler and Navier-Stokes equations. AIAA Paper 2006-304
[107]	Michalak C, Ollivier-Gooch C. 2009. Accuracy preserving limiter for high-order accurate solution of the Euler equations. J. Comput. Phys., 228: 8693-8711
[108]	Nastase C R, Mavriplis D J. 2006. High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys., 213: 330-357
[109]	Oden T J, Carey G F. 1983. Finite Elements: Mathematical Aspects. Vol. IV. Prentice-Hall, Englewood Cliffs, N.J.
[110]	Olivier-Gooch C F. 1997. Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction. J. Comput. Phys., 133: 6-17%
[111]	Park J S, Yoon S H, Kim C. 2010. Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys., 229: 788-812%
[112]	Persson P O, Peraire J. 2006. Sub-cell shock capturing for discontinuous Galerkin methods. AIAA Paper 2006-112
[113]	Qiu J X, Shu C W. 2003. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one dimensional case. J. Comput. Phys., 193: 115-135
[114]	Qiu J X, Shu C W. 2005a. A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput., 27 (3): 995-1013
[115]	Qiu J X, Shu C W. 2005b. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method. Comput. & Fluids, 34: 642-663%
[116]	Qiu J X, Khoo B C, Shu C W. 2006. A numerical study for performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes. J. Comput. Phys., 212: 540-565
[117]	Rasetarinera P, Hussaini M Y. 2001. An efficient implicit discontinuous Galerkin method. J. Comput. Phys., 172: 718-738
[118]	Reed W H, Hill T R. 1973. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory.
[119]	Ren Y X, Liu M, Zhang H X. 2003. A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys., 192 (2): 365-386
[120]	Rider W J, Lowrie R B. 2002. The use of classical Lax-Friedrichs Riemann solvers with discontinuous Galerkin methods. Int J Num Meth Fluids, 40 (3): 479-86
[121]	Roe P L. 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43: 357-372
[122]	Rusanov V V. 1961. Calculation of interaction of non-steady shock waves with obstacles. J Comput Math Phys USSR, 1: 261-279
[123]	Saad Y, Schultz M H. 1986. GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput., 7: 865-884%
[124]	Sharov D, Nakahashi K. 1998. Low speed preconditioning and LU-SGS scheme for 3D viscous flow computations on unstructured grids. AIAA Paper 98-0614
[125]	Shi L, Wang Z J, Fu S, et al. 2012. A P_NP_M-CPR Method for Navier-Stokes Equations. AIAA Paper 2012-460
[126]	Sonar T. 1997. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery accuracy and stencil selection. Comput. Methods Appl. Mech. Eng., 140: 157-181
[127]	Steger J L, Warming R F. 1981. Flux vector splitting of the inviscid gasdynamic equations with application to finite difference methods. J. Comput. Phys., 40: 264-293
[128]	Sun Y, Wang Z J. 2004. Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grid. Int. J. Numer. Methods Fluids, 45 (8): 819-838
[129]	Sun Y, Wang Z J, Liu Y. 2006a. Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow. J. Comput. Phys., 215 (1): 41-58
[130]	Sun Y Z, Wang Z J, Liu Y. 2006b. High-order multi-domain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys., 2: 310-333
[131]	Sun Y Z, Wang Z J, Liu Y and Chen C L. 2007. Efficient implicit LU-SGS algorithm for high-order spectral difference method on unstructured hexahedral grids. AIAA Paper 2007-0313
[132]	Tam C K W, Webb J C. 1993. Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys., 107 (1): 262-81.
[133]	Thareja R R and Stewart J R. 1989. A point implicit unstructured grid solver for the Euler and Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 9: 405-425
[134]	Thompson J F, Soni B K, Weatherill N P. 1999. Handbook of Grid Generation. CRC Press.
[135]	Toulorge T, Desmet W. 2010. Curved boundary treatments for the discontinuous Galerkin method applied to aeroacoustic propagation. AIAA Journal, 48 (2): 479-489%
[136]	Turner M J, Clough R W, Martin H C, et al. 1956. Stiffness and deflection analysis of complex structures. J. Aeronat. Sci., 23 (9)
[137]	Tutkun B, Edis F O. 2012. A GPU application for high-order compact finite difference scheme. Computers & Fluids, 55: 29-35
[138]	van den Abeele K, Lacor C. 2007. An accuracy and stability study of the 2D spectral volume method. J. Comput. Phys., 226: 1007-1026
[139]	van Leer B. 1982. Flux-vector splitting for the Euler equations. Lecture Notes in Physics, 170: 507-512
[140]	Venkatakrishnan V. 1993. On the accuracy of limiters and convergence to steady state solutions. AIAA Paper 93-0880
[141]	Vierendeels J, Riemslagh K, Dick E. 1999. A multigrid semi-implicit line-method for viscous incompressible and low-Mach-number flows on high aspect ratio grids. J. Comput. Phys., 154: 310--341
[142]	Visbal M. 1986. Evaluation of an implicit Navier-Stokes solver for some unsteady separated flows. AIAA Paper 86-1053
[143]	Vos J B, Rizzi A, Darracq D, Hirschel E H. 2002. Navier-Stokes solvers in European aircraft design. Prog. Aerospace Sci., 38: 601-697
[144]	Vries de G, Norrie D H. 1971. The application of the finite element technique to potential flow problems. Transactions, ASME, Series E.; J. Appli. Mech., 38: 243-252%
[145]	Wang L, Mavriplis D J. 2007. Implicit solution of the unsteady Euler equation for high-order accurate discontinuous Galerkin discretizations. J. Comput. Phys., 225: 1994-2005
[146]	Wang Z J. 2002. Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys., 178: 210-251
[147]	Wang Z J, Liu Y. 2002. Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys., 179 (2): 665-697
[148]	Wang Z J, Liu Y. 2004. Spectral (finite) volume method for conservation laws on unstructured grids III: one-dimensional systems and partition optimization. SIAM J. Sci. Comput., 20: 137-157
[149]	Wang Z J, Zhang L, Liu Y. 2004. Spectral finite volume method for conservation laws on unstructured grids IV: extension to two dimensional systems. J. Comput. Phys., 194 (2): 716-741
[150]	Wang Z J, Liu Y. 2005. The spectral difference method for the 2D Euler equations on unstructured grids. AIAA Paper 2005-5112
[151]	Wang Z J, Sun Y, Liang C, et al. 2006. Extension of the SD method to viscous flow on unstructured grids. In: Proceedings of the 4th International Conference on Computational Fluid Dynamics, Gent, Belgium, July 2006.
[152]	Wang Z J. 2007. High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Prog. Aerospace Sci., 43: 1-41
[153]	Wang Z J, Liu Y, May G, et al. 2007. Spectral difference method for unstructured grids II: extension to the Euler equations. SIAM J. Sci. Comput., 32: 45-71
[154]	Wang Z J, Gao H Y. 2009. A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys., 228: 8161-8186
[155]	Wang Z J, Gao H Y, Haga H. 2011. A unifying discontinuous formulation for hybrid meshes. In: Wang Z J, ed. Adaptive High-Order Methods in Computational Fluid Dynamics, World Scientific pp. 423-454%
[156]	Wang Z J, Shi L, Fu S, et al. 2011. A P_NP_M-CPR Framework for Hyperbolic Conservation Laws. AIAA Paper 2011-3227
[157]	Wesseling P. 1990. Multigrid methods in computational fluid dynamics. Z. Angew. Math. Mech., 70: 337-348%
[158]	Wesseling P. 1992. An Introduction to Multigrid Methods. Wiley, Chichester.
[159]	徐守冻. 1993. 求解超/高超声速无黏绕流的自适应有限元方法. [博士论文]. 北京: 北京大学 (Xu S D. 1993. An adaptative finite element scheme for inviscid hypersonic/supersonic flows. [PhD Dissertation]. Beijing: Peking University. (in Chinese))
[160]	Yang M, Wang Z J. 2009. A parameter-free generalized moment limiter for high-order methods on unstructured grids, AIAA Paper 2009-605
[161]	张涵信, 沈孟育. 2003. 计算流体力学差分方法的原理和应用. 北京: 国防工业出版社. (Zhang H X, Shen M Y. 2003. Computational Fluid Dynamics-Fundamentals and Application of Finite Difference Methods. Beijing: National Defense Industrial Press, (in Chinese))
[162]	张涵信. 1986. 无波动、无自由参数的耗散差分格式. 空气动力学学报, 6：143-165 (Zhang H X. 1986. Non-oscillatory and non-free-parameter dissipation difference scheme. Acta Aerodynamica Sinica, 6: 143-165. (in Chinese))
[163]	张来平. 1996a. 非结构网格、矩形/非结构混合网格复杂无粘流场的数值模拟. [博士论文]. 中国空气动力研究与发展中心. (Zhang L P. 1996. Numerical simulation for complex inviscid flow fields on unstructured grids and Cartesian/unstructured hybrid grids. [PhD Dissertation]. CARDC. (in Chinese))
[164]	张来平, 张涵信. 1996b. NND格式在非结构网格中的推广. 力学学报, 28 (2): 135-142. (Zhang L P, Zhang H X. 1996. Development of NND Scheme on Unstructured Grid. Chinese Journal of Theoretical and Applied Mechanics, 28 (2): 135-142. (in Chinese))
[165]	张来平, 刘伟, 贺立新, 等. 2011. 一种新的间断侦测器及其在DGM中的应用, 空气动力学学报, 29 (4): 401-406. (Zhang L. P., Liu W, He LX, et al. 2011. A shock detection method and applications in DGM for hyperbolic conservation laws on unstructured grids. Acta Aerodynamica Sinica, 29 (4): 401-406. (in Chinese))
[166]	Zhang L P, Liu W, He L X, et al. 2012a. A class of hybrid DG/FV methods for conservation laws I: Basic formulation and one-dimensional systems. J. Comput. Phys., 231: 1081-1103
[167]	Zhang L P, Liu W, He L X, et al. 2012b. A class of hybrid DG/FV methods for conservation laws II：Two-dimensional cases. J. Comput. Phys., 231: 1104-1120%
[168]	Zhang L P, Liu W, He L X, et al. 2012. A class of hybrid DG/FV methods for conservation laws III: Two-dimensional Euler equations. Commun. Comput. Phys., 12 (1): 284-314
[169]	Zhang M P, Shu C W. 2005. An analysis and a comparison between the discontinuous Galerkin method and the spectral finite volume methods. Computers & Fluids, 34 (4-5): 581-592
[170]	Zhang S H, Zhang Y T, Shu C W. 2006. Interaction of an oblique shock wave with a pair of parallel vortices: Shock dynamics and mechanism of sound generation. Phys. Fluids, 18: 1-21
[171]	Zhang S H, Shu C W. 2007. A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput., 31 (1/2): 273-305
[172]	Zhang S H, Zhang H X, Shu C W. 2009. Topology structure of shock induced vortex breakdown. J. Fluid Mech., 639: 343-372
[173]	Zhang S H, Jiang S F, Shu C W. 2011. Improvement of convergence to steady state solutions of Euler equations with the WENO schemes. J. Sci. Comput., 47: 216-238%
[174]	Zhong X. 1998. High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition. J. Comput. Phys., 144 (2): 662-709
[175]	Zienkiewicz O C, Cheung Y K. 1965. Finite element method in the solution of field problems. The Engineer, 24: 501-510
[176]	朱刚, 沈孟育. 1995. 跨声速叶栅粘流计算的多级Taylor-Galerkin有限元法. 空气动力学学报, 13 (4): 414-419. (Zhu G, Shen M Y. 1995. Multilevel Taylor-Galerkin finite element method for viscous transonic flow in turbomachinery. Acta Aerodynamica Sinica, 13 (4): 414-419. (in Chinese))
[177]	宗文刚. 2000. 高阶紧致格式及其在复杂流场求解中的应用. [博士论文]. 中国空气动力研究与发展中心. (Zong W G. 2000. High-order compact scheme and its application in complex flow field simulation. [PhD Dissertation]. CARDC. (in Chinese))