Godunov型显式大时间步长格式研究进展

doi:10.7527/S1000-6893.2020.23575

综述

本期目录 | 过刊浏览 | 高级检索

前一篇 | 后一篇

Godunov型显式大时间步长格式研究进展

钱战森^1,2

1. 中国航空工业空气动力研究院, 沈阳 110034;
2. 高速高雷诺数气动力航空科技重点实验室, 沈阳 110034

收稿日期:2019-10-14 修回日期:2020-01-20 出版日期:2020-07-15 发布日期:2020-02-13
通讯作者: 钱战森 E-mail:qianzs@avicari.com.cn
基金资助:
国家自然科学基金（11672280，11202199）

Research progress of Godunov type explicit large time step scheme

QIAN Zhansen^1,2

1. AVIC Aerodynamics Research Institute, Shenyang 110034, China;
2. Aviation Key Laboratory of Science and Technology on High Speed and High Reynolds Number Aerodynamic Force Research, Shenyang 110034, China

Received:2019-10-14 Revised:2020-01-20 Online:2020-07-15 Published:2020-02-13
Supported by:
National Natural Science Foundation of China (11672280,11202199)

摘要/Abstract

摘要： 综述了Godunov型显式大时间步长格式的研究进展。首先介绍了显式大时间步长格式的概念、分类和优势。然后重点阐述了Godunov型显式大时间步长格式的构造方法、高阶精度推广方法、多维问题推广方法和收敛特性、分辨率及计算效率等性能，展示了其在典型问题中的应用和验证。最后给出了Godunov型显式大时间步长格式研究进一步可能的发展方向。

关键词: 大时间步长格式, Godunov型格式, 高效算法, 格式稳定性, 显式格式

Abstract: The research progress of the Godunov type explicit large time step scheme is reviewed. Firstly, the concept, classifications and advantages of explicit large time step scheme are introduced. Then, followed by its construction methods, higher order accuracy extension approaches, multi-dimension generalization methods, and characteristics analysis including stability, resolution and computational efficiency. Finally, further development direction of the Godunov type explicit large time step scheme is proposed.

Key words: large time step schemes, Godunov type schemes, highly efficient algorithm, scheme stability, explicit schemes

中图分类号:

V211.3

钱战森. Godunov型显式大时间步长格式研究进展[J]. 航空学报, 2020, 41(7): 23575-023575.

QIAN Zhansen. Research progress of Godunov type explicit large time step scheme[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020, 41(7): 23575-023575.

参考文献 118

[1]	朱自强. 应用计算流体力学[M]. 北京:北京航空航天大学出版社, 1998. ZHU Z Q. Applied computational fluid dynamics[M]. Beijing:Beihang University Press, 1998(in Chinese).
[2]	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:High Education Press, 2002(in Chinese).
[3]	吴子牛. 计算流体力学基本原理[M]. 北京:科学出版社, 2002. WU Z N. Basic principles of computational fluid dynamics[M]. Beijing:Science Press, 2002(in Chinese).
[4]	张涵信, 沈孟育. 计算流体力学——差分方法的原理和应用[M]. 北京:国防工业出版社, 2003. ZHANG H X, SHEN M Y. Computational fluid dynamics——Theory and application of finite difference method[M]. Beijing:National Defense Industry Press, 2003(in Chinese).
[5]	任玉新, 陈海昕. 计算流体力学基础[M]. 北京:清华大学出版社, 2006. REN Y X, CHEN H X. Foundations of computational fluid dynamics[M]. Beijing:Tsinghua University Press, 2006(in Chinese).
[6]	阎超. 计算流体力学方法及应用[M]. 北京:北京航空航天大学出版社, 2006. YAN C. Computational fluid dynamics methods and its application[M]. Beijing:Beihang University Press, 2006(in Chinese).
[7]	ROACHE P J. Computational fluid dynamics[M]. Socorro:Hermosa Publisher, 1972.
[8]	FLECTCHER C A J. Computational techniques for fluid dynamics[M]. New York:Spring-Verlag, 1988.
[9]	ANDERSON J D. Computational fluid dynamics:Basics with applications[M]. New York:McGraw-Hill, 1995.
[10]	PEYRET R. Handbook of computational fluid mechanics[M]. Pittsburgh:Academic Press, 1996.
[11]	TORO E F. Riemann solvers and numerical methods for fluid dynamics:A practical introduction[M]. Berlin:Springer, 1997.
[12]	LEVEQUE R J. Finite volume methods for hyperbolic problems[M]. Combridge:Combridge University Press, 2002.
[13]	黄志澄. 高超声速飞行器空气动力学[M]. 北京:国防工业出版社, 1995. HUANG Z C. Hypersonic aircraft aerodynamics[M]. Beijing:National Defense Industry Press, 1995(in Chinese).
[14]	张兆顺, 崔桂香, 许晓春. 湍流理论与模拟[M]. 北京:清华大学出版社, 2005. ZHANG Z S, CUI G X, XU C X. Turbulence theory and simulation[M]. Beijing:Tsinghua University Press, 2005(in Chinese).
[15]	SCHETZ J A. Aerodynamics of high-speed trains[J]. Annual Review of Fluid Mechanics, 2001, 53(2):371-414.
[16]	郑晓静. 风沙运动的沙粒带电机理及其影响的研究进展[J]. 力学进展, 2004, 34(1):77-86. ZHENG X J. Advances in investigation on electrification of wind-blown sands and its effects[J]. Advances in Mechanics, 2004, 34(1):77-86(in Chinese).
[17]	李劲菁. 基于高阶熵条件格式的Euler方程与Navier-Stokes方程混合算法[D]. 北京:北京航空航天大学, 2002. LI J J. On the hybrid algorithm of Euler and Navier-Stokes equations based on high-order entropy condition scheme[D]. Beijing:Beihang Univeristy, 2002(in Chinese).
[18]	THOMAS J W. Numerical partial differential equations[M]. New York:Springer-Verlag, 1995.
[19]	CROCCO L. A suggestion for the numerical solution of the steady Navier-Stokes equations[J]. AIAA Journal, 1965, 3(10):1824-1832.
[20]	MORETTI G, ABBETT M. A time-dependent computational method for blunt body flows[J]. AIAA Journal, 1966, 4(12):2136-2141.
[21]	水鸿寿. 一维流体力学差分方法[M]. 北京:国防工业出版社, 1998. SHUI H S. One dimensional fluid mechanics finite difference method[M]. Beijing:National Defense Industry Press, 1998(in Chinese).
[22]	沈荣华, 冯果忱. 微分方程数值解法[M]. 北京:人民教育出版社,1980. SHEN R H, FENG G C. Numerical methods of partial differential equation[M]. Beijing:People's Education Press, 1980(in Chinese).
[23]	BEAM R M, WARMING R F. An implicit finite-difference algorithm for hyperbolic system in conservation law form[J]. Journal of Computational Physics, 1976, 22:87-109.
[24]	MACCORMACK R W. A numerical method for solving the equations of compressible viscous flow:AIAA-1981-0110[R]. Reston:AIAA, 1981.
[25]	PULLIAM T H, STEGER J L. Recent improvements in efficiency, accuracy, and convergence for implicit approximate factorization algorithms:AIAA-1985-0360[R]. Reston:AIAA,1985.
[26]	PULLIAM T H, CHAUSSEE D S. A diagonal form of an implicit approximate factorization algorithm[J]. Journal of Computational Physics, 1981, 39:347-363.
[27]	YOON S, JAMESON A. Lower-upper symmetric Gauss-Sediel method for the Euler and Navier-Stokes equations[J]. AIAA Journal, 1988, 26(9):1025-1026.
[28]	LUO H, BAUM J D, LOHNER R. Matrix-free implicit method for compressible flow on unstructured grids[J]. Journal of Computational Physics, 1998,146:664-690.
[29]	PUEYO A, ZINGG D W. Efficient Newton-Krylov solver for aerodynamic computations[J]. AIAA Journal, 1998, 36(11):1991-1997.
[30]	JAMESON A. Time dependent caculations using multigrid with application to unsteady flows past airfoils and wings:AIAA-1991-1596[R]. Reston:AIAA, 1991.
[31]	GODONUV S K. A finite difference method for the computation of discontinuous solutions of the equations of fluids dynamics[J]. Matematichestki Sbornik, 1959, 47(89):271-306(in Russian).
[32]	BORIS J P, BOOK D L. Flux corrected transport Ⅰ. SHASTA, a fluid transport algorithm that works[J]. Journal of Computational Physics, 1973, 11:25-40.
[33]	HARTEN A, ZWAS G. Self-adjusting hybrid schemes for shock computations[J]. Journal of Computational Physics, 1972, 9:568-583.
[34]	HARTEN A. High resolution schemes for hypersonic conservation laws[J]. Journal of Computational Physics, 1983, 49:357-393.
[35]	JAMESON A, SCHMIDT W, TURKEL E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes:AIAA-1981-1259[R]. Reston:AIAA, 1981.
[36]	VAN LEER B. Towards the ultimate conservation difference scheme:Ⅱ. Monotonicity and conservation combined in a second order scheme[J]. Journal of Computational Physics, 1974, 14:361-370.
[37]	VAN LEER B. Towards the ultimate conservation difference scheme:Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow[J]. Journal of Computational Physics, 1977, 23:263-275.
[38]	VAN LEER B. Towards the ultimate conservation difference scheme:Ⅳ. A new approach to numerical convection[J]. Journal of Computational Physics, 1977, 23:276-299.
[39]	VAN LEER B. Towards the ultimate conservation difference scheme:Ⅴ. A second-order sequel to Godunov's method[J]. Journal of Computational Physics, 1979, 3:101-136.
[40]	COLLELA P, WOODWARD P. The piecewise parabolic method for gas-dynamical simulations[J]. Journal of Computational Physics, 1984, 54:264-289.
[41]	HARTEN A, ENGQUIST B, OSHER S, et al. Uniformly high-order accurate essentially non-oscillatory schemes Ⅲ[J]. Journal of Computational Physics, 1987, 71:231-303.
[42]	LIU X D, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics, 1994, 115:200-212.
[43]	JIANG G, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126:208-228.
[44]	LELE S K. Compact finite difference schemes with spectral-like resolution[J]. Journal of Computational Physics, 1992, 103:16-42.
[45]	FU D X, MA Y W. A high order accurate difference scheme for complex flow[J]. Journal of Computational Physics, 1997, 134:1-15.
[46]	ADAMS N A, SHARIFF K. A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems[J]. Journal of Computational Physics, 1996, 127:27-51.
[47]	DENG X G, MAEKAWA H. Compact high-order accurate nonlinear schemes[J]. Journal of Computational Physics, 1997, 130:77-91.
[48]	DENG X G, ZHANG H X. Developing high-order accurate nonlinear schemes[J]. Journal of Computational Physics, 2000, 165:22-44.
[49]	PIROZZOLI S. Conservative hybrid compact-WENO schemes for shock-turbulence interaction[J]. Journal of Computational Physics, 2002, 178:81-117.
[50]	REN Y X, LIU M, ZHANG H X. A characteristic hyprid compact-WENO scheme for solving hyperbolic conservation laws[J]. Journal of Computational Physics, 2003, 192:365-386.
[51]	YEE H C, SJOGREEN B. Nonlinear filtering in compact high order schemes for ideal and non-ideal MHD equations[J]. Journal of Scientific Computing, 2006, 27:507-521.
[52]	YEE H C, SJOGREEN B. Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems[J]. Journal of Computational Physics, 2007, 225:910-934.
[53]	ZHANG S H, JIANG S F, SHU C W. Development of nonlinear weighted compact schemes with increasingly higher order accuracy[J]. Journal of Computational Physics, 2008, 227:7294-7321.
[54]	ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics, 1981, 43:357-372.
[55]	ENGGUIST B, OSHER S. One-side difference approximations for nonlinear conservation laws[J]. Mathematics of Computation, 1981, 36(154):321-351.
[56]	OSHER S, SOLOMON F. Upwind difference schemes for hyperbolic conservation laws[J]. Mathematics of Computation, 1982, 38:339-374.
[57]	HARTEN A, LAX P D, VON LEER B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J]. SIAM Review, 1983, 25(1):35-61.
[58]	TORO E F, SPRUCE M, SPEARES W. Restoration of the contact surface in the HLL-Riemann solver[J]. Shock Waves, 1994, 4:25-34.
[59]	STEGER J L, WARMING B. Flux vector splitting of the inviscid gasdynamic equations with applications to finite difference methods[J]. Journal of Computational Physics, 1981, 40:263-293.
[60]	VAN LEER B. Flux-vector splitting for the equations:NASA TR 82-30[R]. Washington, D.C.:NASA Langley Reaserch Center, 1982.
[61]	LIOU M S, STEFFEN C J. A new flux splitting scheme[J]. Journal of Computational Physics, 1993, 107:23-39.
[62]	LIOU M S. A sequel to AUSM:AUSM+[J]. Journal of Computational Physics, 1996, 129:364-382.
[63]	WADA Y, LIOU M S. An accurate and robust flux splitting scheme for shock and contact discontinuities[J]. SIAM Journal on Scientific and Statistical Computing, 1997, 18:633-657.
[64]	LIOU M S. Mass flux schemes and connection to shock instability[J]. Journal of Computational Physics, 2000, 160:623-648.
[65]	LIOU M S. Ten years in the making-AUSM family:AIAA-2001-2521[R]. Reston:AIAA, 2001.
[66]	TATSUMI S, MARTINELLI L, JAMESON A. Design, implementation, and validation of flux limited schemes for the solution of the compressible Navier-Stokes equations:AIAA-1994-0647[R]. Reston:AIAA, 1994.
[67]	张涵信. 无波动、无自由参数的耗散差分格式[J]. 空气动力学学报, 1988, 6(2):143-165. ZHANG H X. Non-oscillation, non-free parameters dissipative finite difference scheme[J]. Acta Aerodynamica Sinica, 1988, 6(2):143-165(in Chinese).
[68]	张涵信. 无波动、无自由参数、耗散的隐式差分格式[J]. 应用数学与力学, 1991, 12(1):97-100. ZHANG H X. Non-oscillation, non-free parameters and dissipation implicit finite difference scheme[J]. Applied Mathematics and Mechanics, 1991, 12(1):97-100(in Chinese).
[69]	LAX P D, WENDROFF B. Difference schemes for hyperbolic equations with high order of accuracy[J]. Communications Pure and Applied Mathematics, 1964, 17:381-393.
[70]	MACCORMACK R W. The effect of viscosity in hypervelocity impact cratering:AIAA-1969-0354[R]. Reston:AIAA, 1969.
[71]	NESSYAHU H, TADMOR E. Non-oscillatory central differencing for hyperbolic conservation laws[J]. Journal of Computational Physics, 1990, 87:408-463
[72]	JIANG G S, LEVY D, LIN C T, et al. High-resolution nonoscillatory central schemes with nonstaggered grid for hyperbolic conservation laws[J]. SIAM Journal on Numerical Analysis, 1998, 35(6):2147-2168.
[73]	LEVY D, PUPPO G, RUSSO G. Central WENO schemes for hyperbolic systems of conservation laws[J]. Mathematical Modelling and Numerical Aanalysis, 1999, 33(3):547-571.
[74]	LEVY D, PUPPO G, RUSSO G. Compact central WENO schemes for multidimensional conservation laws[J]. SIAM Journal on Scientific Computing, 2000, 22(2):656-672.
[75]	LAX P D. Hyperbolic systems of conservation laws[J]. Communications Pure and Applied Mathematics, 1960, 13:217-237.
[76]	DAVIS S. A rotationally biased upwind difference scheme for the Euler equations[J]. Journal of Computational Physics, 1984, 56:65-92.
[77]	REN Y X. A robust shock-capturing scheme based on rotated Riemann solvers[J]. Computers & Fluids, 2003, 32:1379-1403.
[78]	HIROAKI N, KEIICHI K. Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers[J]. Journal of Computational Physics, 2008, 227:2560-2581.
[79]	GHISLAIN T, PASCALIN T K, YVES B. An accurate shock-capturing scheme based on rotatedhybrid Riemann solver:AUFSRR scheme[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2016, 26(5):1310-1327.
[80]	LEVEQUE R J. Large time-step shock capture techniques for scalar conservation laws[J]. SIAM Journal on Numerical Analysis, 1982, 19:1091-1109.
[81]	LEVEQUE R J. Convergence of a large time step generalization of godunov's method for conservation laws[J]. Communications on Pure and Applied Mathematics, 1984, 37(4):463-477.
[82]	LEVEQUE R J. A large time step generalization of Godunov's method for systems of conservation laws[J]. SIAM Journal on Numerical Analysis, 1985, 22(6):1051-1073.
[83]	GUINOT V. The time-line interpolation method for large-time-step Godunov-type schemes[J]. Journal of Computational Physics, 2002, 177:394-417.
[84]	QIAN Z S, LEE C H. A class of large time step Godunov schemes for hyperbolic conservation laws and applications[J]. Journal of Computational Physics, 2011, 230(19):7418-7440.
[85]	钱战森. 大时间步长、高分辨率差分格式研究及其应用[D]. 北京:北京航空航天大学,2011. QIAN Z S. On large time step, high resolution finite difference scheme and its application[D]. Beijing:Beihang Univeristy, 2011(in Chinese).
[86]	DONG H T, LIU F J. Large time step wave adding scheme for systems of hyperbolic conservation laws[J]. Journal of Computational Physics, 2018, 374:331-360.
[87]	HARTEN A. On a large time-step high resolution scheme[J]. Mathematics of Computation, 1986, 46(174):379-399.
[88]	董海涛, 李椿萱. 快速大时间步长熵条件格式的分辨率研究[J]. 北京航空航天大学学报, 2003, 29(11):1011-1016. DONG H T, LEE C H. Researches on the resolution of fast large time step entropy condition scheme[J]. Journal of Beijing University of Aeronautics and Astronautics,2003, 29(11):1011-1016(in Chinese).
[89]	QIAN Z S, LEE C H. On large time step TVD scheme for hyperbolic conservation laws and its efficiency evaluation[J]. Journal of Computational Physics, 2012, 231:7415-7430.
[90]	PULLIAM T H, STEGER L. Recent improvements in efficiency, accuracy, and convergence for implicit approximate factorization algorithms:AIAA-1985-0360[R]. Reston:AIAA, 1985.
[91]	VENKARAKRISHNAN V, JAMESON A. Computation of unsteady transonic flows by the solution of Euler equations[J]. AIAA Journal, 1988, 26(8):974-981.
[92]	JORGENSON P, CHIMA R. An unconditionally stable Runge-Kutta method for unsteady flows:AIAA-1989-0205[R]. Reston:AIAA, 1989.
[93]	CHADERJIAN N M, GURUSWAMY G P. Unsteady transonic Navier-Stokes computations for an oscillating wing using single and multiple zones:AIAA-1990-0313[R]. Reston:AIAA, 1990.
[94]	MOITRA A. Enthalpy damping for high Mach number Euler solutions[J]. AIAA Journal, 1992, 30(2):300-301.
[95]	JAMESON A, YOON S. Multigrid solution of the Euler equations using implicit schemes:AIAA-1985-0293[R]. Reston:AIAA, 1985.
[96]	COURANT R, FRIEDRICHS K O. Supersonic flow and shock waves[M]. Berlin:Springer, 1999.
[97]	LINDQVIST S, AURSAND P, FLATTEN T, et al. Large time step TVD schemes for hypersonic conservation laws[J]. SIAM Journal on Numerical Analysis, 2016, 54(5):2775-2798
[98]	PREBEG M, FLATTEN T, MULLER B. Large time step HLL and HLLC schemes[M]//ESAIM:Mathematical Modelling and Numerical Analysis, 2018, 52:1239-1260.
[99]	EINFELDT B, MUNZ C D, ROE P L, et al. On Godunov-type methods near low densities[J]. Journal of Computational Physics, 1991, 92(2):273-295.
[100]	BILLETT S J, TORO E F. On waf-type schemes for multidimensional hyperbolic conservation laws[J]. Journal of Computational Physics, 1997, 130:1-24.
[101]	STRANG G. On the construction and comparison of difference schemes[J]. SIAM Journal on Numerical Analysis, 1968, 5:506-517.
[102]	ANDERSON W K, THOMAS J L, VAN LEER B. Comparison of finite volume flux vector splittings for the Euler equations[J]. AIAA Journal, 1986, 24(9):1453-1460.
[103]	DADONE A, GROSSMAN B. Surface boundary conditions for the numerical solution of the Euler equations:AIAA-1993-3334[R]. Reston:AIAA, 1993.
[104]	LEVEQUE R J. Convergence of a large time step generalization of Godunov's method for conservation laws[J]. Communications on Pure and Applied Mathematics, 1984, 37(4):463-477.
[105]	JAMESON A, LAX P D. Conditions for the construction of multi-point total variationl diminishing difference sche-mes[J]. Applied Numerical Mathematics, 1986, 2(3-5):335-345.
[106]	JAMESON A, LAX P D. Corrigendum:Conditions for the construction of multi-point total variation diminishing difference schemes[J]. Applied Numerical Mathematics, 1987, 3(3):289.
[107]	WANG J, WARNECKE G. On entropy consistency of large time step schemes I. The Godunov and Glimm schemes[J]. SIAM Journal on Numerical Analysis, 1993, 30(5):1229-1251, 1993.
[108]	WANG J, WARNECKE G. On entropy consistency of large time step schemes II. Approximate Riemann solvers[J]. SIAM Journal on Numerical Analysis, 1993, 30(5):1252-1267.
[109]	WANG J, WEN H, ZHOU T. On large time step Godunov scheme for hyperbolic conservation laws[J]. Communications in Mathematical Sciences, 2004, 2(3):477-495.
[110]	TANG H, WARNECKE G. A note on (2K+1)-point conservative monotone schemes[J]. ESAIM:Mathematical Modelling and Numerical Analysis, 2004, 38(2):345-357.
[111]	MURILLO J, NAVARRO P G, BRUFAU P, et al. Extension of a finite volume method to large time steps (CFL>1):Application to shallow water flows[J]. International Journal for Numerical Methods in Fluids, 2006, 50:63-102.
[112]	HERNANDEZ M M, NAVARRO P G, MURILLO J. A large time step 1D upwind explicit scheme (CFL>1):Application to shallow water equations[J]. Journal of Computational Physics, 2012, 231:6532-6557.
[113]	HERNANDEZ M M, HUBBARD M E, NAVARRO P G.A 2D extension of a large time step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries[J]. Journal of Computational Physics, 2014, 263:303-327.
[114]	HERNANDEZ M M, LACASTA A, MURILLO J, et al. A large time step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws:Application to the homogeneous shallow water equations[J]. Applied Mathematical Modelling, 2017, 47:294-317.
[115]	XU R, ZHONG D, WU B, et al. A large time step Godunov scheme for free-surface shallow water equations[J]. Chinese Science Bulletin, 2014, 59:2534-2540.
[116]	THOMPSON R J, MOELLER T. A discontinuous wave-in-cell numerical scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2015, 299:404-428.
[117]	SOD G A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J]. Journal of Computational Physics, 1978, 27:1-31.
[118]	YEE H C. Construction of explicit and implicit symmetric TVD schemes and their applications[J]. Journal of Computational Physics, 1987, 68:151-179.

E-mail：hkxb@buaa.edu.cn

关于我们

期刊社服务

专业学科

封面文章

友情链接

主管单位：中国科学技术协会主办单位：中国航空学会北京航空航天大学

Godunov型显式大时间步长格式研究进展

Research progress of Godunov type explicit large time step scheme

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献 118

相关文章 2

编辑推荐

Metrics

本文评价

[1]	辛冀, 李攀, 陈仁良. 基于三阶显式格式的旋翼时间步进自由尾迹计算与验证[J]. 航空学报, 2013, 34(11): 2452-2463.
[2]	杨弘炜;李椿萱. 模拟三维粘性流动的显式多重网格算法[J]. 航空学报, 1994, 15(11): 1283-1290.