Arbitrary Lagrangian-Eulerian方法及其关键技术研究
|
摘要
在流体大变形问题的数值模拟中，Arbitrary Lagrangian-Eulerian（ALE）方法皆具Euler和Lagrange方法的优点，是目前国内外重点研究与广泛应用的方法之一。针对复杂流体大变形问题的求解，目前ALE方法研究的关键工作主要包括高精度Lagrange有限体积格式的构造，计算网格变形的网格重构方法研究，以及高精度守恒重映方法的研究。本论文针对流体大变形问题的高精度ALE方法数值模拟的研究课题，分别构造了一类交错网格上的高精度ENO型Lagrange有限体积格式和一类两个格心型高精度Lagrange有限体积格式，并结合任意多边形相交技术与网格贡献法思想提出了两类适用于任意网格的守恒重映方法，并成功地将自适应网格技术引入到ALE方法中，实现了一类自适应的ALE方法。主要研究内容有：
（1）使用最小二乘多项式重构（LSM）和含有特征分解的ENO多项式重构方法，结合构造时空高精度格式的思想，构造得到了一类两个结构网格下格心型高精度Lagrange有限体积格式。
（2）使用结构网格下的ENO重构多项式技术，推广了四边形结构网格下的一阶有限体积格式，构造得到了交错网格上的ENO型高精度Lagrange有限体积格式。
（3）通过研究任意多边形相交计算问题，实现了一类任意两个多边形相交算法，构造了一类基于ENO重构思想的高精度守恒重映方法。
（4）在分析了二阶保号守恒重映方法的基础上，采用重构多项式的方法代替原算法中的误差补偿方法，利用网格贡献法思想，构造了ENO近似积分守恒重映方法和LSM近似积分两个高精度守恒重映方法。
（5）将高精度Lagrange有限体积格式和高精度守恒重映方法耦合在一起，实现了高精度ALE方法的数值模拟。通过一系列数值算例验证了算法的高精度和可行性。在此基础上，结合有效的自适应网格重构方法，成功的构造了一类自适应的ALE方法。数值模拟的结果表明自适应ALE方法有效的提高了激波和接触间断处数值解的分辨率。
In the research of simulation for problems with large distortion in fluid dynamics, one of popular numerical algorithms is so called Arbitrary Lagrangian-Eulerian (ALE) method. For the reason of concerning with large mesh distortion, the research of this kind of method mainly focuses on constructing high order schemes in Lagrangian formulism, meshes regenerate for grid distortion and conservative remapping algorithms for transferring physical quantities between two computational meshes. In this paper, a class of high order ENO Lagrangian finite volume schemes on the staggered and a class of two cell-centered high order Lagrangian finite volume schemes have been developed. Combining with the ENO interpolation, two kinds of conservative remapping algorithms have been developed for arbitrary mesh systems based on computing intersections of two polygons and cell donor method. It primarily concerns five following aspects:
Firstly, a class of two cell-centered high order finite volume schemes is developed on structured grids in Lagrangian formulism. By using high order ENO reconstruction with Roe - type characteristic decomposition of Least Squares method (LSM) on structured grids, a high order ENO of LSM Lagrangian finite volume scheme both on space and time is developed .
Secondly, a high order finite volume scheme is developed on the staggered mesh in Lagrangian formulism. By using ENO interpolating polynomials, a high order ENO finite volume scheme has been constructed through modifying a first order finite volume scheme on rectangular mesh.
Thirdly, a new kind of high order conservative remapping algorithm is constructed. Through researching the problem of finding cell intersections, a new kind of conservative remapping algorithm has been constructed by using high order ENO reconstruction.
Fourthly, a kind of approximate integration conservative remapping algorithm is constructed. Through analyzing a second order sign preserving conservative remapping method, two conservative remapping algorithms have been constructed by using the ENO reconstruction and Least Squares method to take the place of the positivity preserving error compensation algorithm.
At last, combining the high order finite volume schemes and the efficient remapping algorithms, we succeed to develop ALE method for numerical computations. Testing with a set of reliable numerical examples, it is shown that the ALE algorithm proposed presently be effective. Furthermore, coupling adaptive mesh generation method with our ALE method, a new kind of adaptive ALE method is developed. One and two dimensional numerical examples are presented to demonstrate the performance of the schemes in terms of accuracy and resolution for contact discontinuities and shock regions.
（1）使用最小二乘多项式重构（LSM）和含有特征分解的ENO多项式重构方法，结合构造时空高精度格式的思想，构造得到了一类两个结构网格下格心型高精度Lagrange有限体积格式。
（2）使用结构网格下的ENO重构多项式技术，推广了四边形结构网格下的一阶有限体积格式，构造得到了交错网格上的ENO型高精度Lagrange有限体积格式。
（3）通过研究任意多边形相交计算问题，实现了一类任意两个多边形相交算法，构造了一类基于ENO重构思想的高精度守恒重映方法。
（4）在分析了二阶保号守恒重映方法的基础上，采用重构多项式的方法代替原算法中的误差补偿方法，利用网格贡献法思想，构造了ENO近似积分守恒重映方法和LSM近似积分两个高精度守恒重映方法。
（5）将高精度Lagrange有限体积格式和高精度守恒重映方法耦合在一起，实现了高精度ALE方法的数值模拟。通过一系列数值算例验证了算法的高精度和可行性。在此基础上，结合有效的自适应网格重构方法，成功的构造了一类自适应的ALE方法。数值模拟的结果表明自适应ALE方法有效的提高了激波和接触间断处数值解的分辨率。
In the research of simulation for problems with large distortion in fluid dynamics, one of popular numerical algorithms is so called Arbitrary Lagrangian-Eulerian (ALE) method. For the reason of concerning with large mesh distortion, the research of this kind of method mainly focuses on constructing high order schemes in Lagrangian formulism, meshes regenerate for grid distortion and conservative remapping algorithms for transferring physical quantities between two computational meshes. In this paper, a class of high order ENO Lagrangian finite volume schemes on the staggered and a class of two cell-centered high order Lagrangian finite volume schemes have been developed. Combining with the ENO interpolation, two kinds of conservative remapping algorithms have been developed for arbitrary mesh systems based on computing intersections of two polygons and cell donor method. It primarily concerns five following aspects:
Firstly, a class of two cell-centered high order finite volume schemes is developed on structured grids in Lagrangian formulism. By using high order ENO reconstruction with Roe - type characteristic decomposition of Least Squares method (LSM) on structured grids, a high order ENO of LSM Lagrangian finite volume scheme both on space and time is developed .
Secondly, a high order finite volume scheme is developed on the staggered mesh in Lagrangian formulism. By using ENO interpolating polynomials, a high order ENO finite volume scheme has been constructed through modifying a first order finite volume scheme on rectangular mesh.
Thirdly, a new kind of high order conservative remapping algorithm is constructed. Through researching the problem of finding cell intersections, a new kind of conservative remapping algorithm has been constructed by using high order ENO reconstruction.
Fourthly, a kind of approximate integration conservative remapping algorithm is constructed. Through analyzing a second order sign preserving conservative remapping method, two conservative remapping algorithms have been constructed by using the ENO reconstruction and Least Squares method to take the place of the positivity preserving error compensation algorithm.
At last, combining the high order finite volume schemes and the efficient remapping algorithms, we succeed to develop ALE method for numerical computations. Testing with a set of reliable numerical examples, it is shown that the ALE algorithm proposed presently be effective. Furthermore, coupling adaptive mesh generation method with our ALE method, a new kind of adaptive ALE method is developed. One and two dimensional numerical examples are presented to demonstrate the performance of the schemes in terms of accuracy and resolution for contact discontinuities and shock regions.
引文
[1]Hirt C W, Amsden A A and Cook J L. An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 1974, 14, 76-85.
[2]Von Neumann J and Richtmyer R D. A method for the numerical calculations of hydrodynamics shocks. J. Appl. Phys. 1950, 21: 232.
[3]Kolsky H G. A method for the numerical solution of transient hydrodynamics shock problem in two space dimensions. Report LA-1867, Los Alamos Sci. Lab. 1955.
[4]Browne P L. REZONE, A proposal for accomplishing rezoning in two- dimensional Lagrangian hydrodynamics problems. Report LA-3455-MS, Los Alamos Sci. Lab. 1966.
[5]Liu W K, Chang H and Belytscko T. Arbitrary Lagrangian-Eulerian Petrov- Galerkin finite elements for nonlinear continue. Comput. Meth. Appl. Mech. Eng. 1988, 68(3):259-310.
[6]Nomura T. ALE finite element computations of fluid-structure interaction problems. Comput. Meth. Appl. Mech. Eng. 1994, 112(1/4):291-308.
[7]Brooks A N and Hughes T R. Streamline upwind Petrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 1982, 32(1-3):199- 259.
[8]Babu S S and Bhattacharyya S K. Finite element analysis of fiuid-structure interaction effect on liquid retaining structures due to sloshing. Computers & Structure. 1996, 59(6):1105-1171.
[9]Liu W K and Chen J S. Adaptive ALE finite element with particular reference to external work rate on frictionalinterface. Comput. Meth. Appl. Mech. Eng. 1991, 93:189-216.
[10]Huerta A and Liu W K. Viscous flow with large free surface motion. Comput. Meth. Appl. Mech. Eng. 1988, 69(3):277-324.
[11]Chorin A J. Numerical solution of the Navier-Stokes equation. Math. Comput. 1968, 22(104):745-762.
[12]Ushijima S. Three dimensional arbitrary Lagrangian-Eulerian numerical prediction methods for nonlinear freesurface oscillation. Int. J. Numer. Meth. in Fluids. 1998, 26:605-623.
[13]Benson D J. Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Meth. Appl. Mech. Eng. 1992, 99:235-394.
[14]Darlington R M, et al. A study of ALE simulations of Rayleigh-Taylor instability. Computer Physics Communications. 2002, 135(1): 58-73.
[15]Peterkin R E, et al. Transport of magnetic flux in an arbitrary coordinate ALE code. J. Comput. Phys. 1998, 140 (1):148-171.
[16]Shestakov A L, et al. "The ICF3D Code". Comp. Meth. Appl. Mech. Eng. 2000, 187: 181-201.
[17]Smith R W. AUSM (ALE): a geometrically conservative arbitrary Langrangian -Eulerian flux splitting scheme. J. Comput. Phys. 1999, 150 (1):268-286.
[18]Arber T D, et al. A staggered grid, Lagrangian- Eulerian remap code for 3-D MHD simulations. J. Comput. Phys. 2001, 171(1):151-181.
[19]Rieben R N, et al. An arbitrary Lagrangian Eulerian discretization of MHD on 3D unstructured grids. J. Comput. Phys. 2007,226(1):534-570.
[20]Pierce T and Rodrigue G. A parallel two-sided contact algorithm in ALE3D. Comput. Methods Appl. Mech. Eng. 2005, 194 (27):3127–3146.
[21]Nichols A, et al. Coupled thermal/chemcial/mechanical modeling of energetic materials in ALE3D. UCRL-JC-124706. Lawrence Livermore National Laboratory. 1996.
[22]Couch R, et al. Application of ALE techniques to metal forming simulations. UCRL-JC-114851. Lawrence Livermore National Laboratory. 1993.
[23]Mcclell R, et al. ALE3D model predictions and materials charcterization for the cookoff response of PBXN-109. UCRL-JC-145756. Lawrence Livermore National Laboratory. 2002.
[24]Couch R and Faux D. Simulation of underwater benchmark experiments with ALE3D. UCRL-JC-123819. Lawrence Livermore National Laboratory. 1997.
[25] Stevens D. Compressible multiphase flows in an ALE framework. UCRL-JC -201373. Lawrence Livermore National Laboratory. 2003.
[26]Aro C, Dube E, Futral W. Coupled mechanical/heat transfer simulation on MPP platforms using a finite element linear solver interface. UCRL-JC- 131746. Lawrence Livermore National Laboratory. 1999.
[27]Tipton R. A 2D lagrange MHD code. UCRL-JC-94277. Lawrence Livermore National Laboratory.
[28]Tipton R. Modeling flux compression generators with a 2D ALE code. UCRL -JC-99900. Lawrence Livermore National Laboratory.
[29]阚明先等.衬套内爆ALE方法二维MHD数值模拟.四川大学学报. 2007, 44 (1): 91-96.
[30]Margolin L G. Introduction to“An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds”. J. Comput. Phys. 1997, 135:198-202.
[31]Goad W B. WAT: A numerical method for two-dimensional unsteady fluid flow. LAMS-2365. Los Alamos Sci. Lab. 1960.
[32]Herrmann W. Comparison of finite difference expressions used in Lagrangian fluid flow calculations. AFWL-TR-64-104. Air Force Weapon Lab. 1964.
[33]Wilkins M L. Calculation of elastic-plastic flow. Methods in Computational Physics. 1964, 3:211.
[34]Schulz W D. Two-dimensional Lagrangian hydrodynamic difference equations. Methods in Computational Physics. 1964, 3:1.
[35]Maenchen G and Sack S. The tensor code. Methods in Computational Physics. 1964, 3:181.
[36]Browne P L and Wallick K B. The reduction of mesh tangling in two- dimensional Lagrangian hydrodynamics codes by the use of viscosity, artificial viscosity and TTS, for long, thin zones. LA-4740-MS. Los Alamos Sci. Lab. 1971.
[37]Amsden A A, et al. SALE-3D:A simplified ALE computer program for fluid flow at all flow speed. Los Alamos Scientic Laboratory. 1980.
[38]Amsden A A, et al. SALE-3D: A simplified ALE computer program for calculating three-dimensional fluid flow. Los Alamos Scientic Laboratory. 1981.
[39]Benson D J. An efficient, accurate, simple ALE method for nonlinear finite element programs. Comput. Meth. Appl. Mech. Eng. 1989, 72:305-350.
[40]Benson D J. Vectorization techniques for explicit arbitrary Lagrangian- Eulerian calculations. Comput. Meth. Appl. Mech. Eng. 1992, 96:303-328.
[41]Pember R, et al. A comparison of staggered-mesh Lagrange plus remap and cell-centered direct Eulerian Godunov schemes for Eulerian shock hydrodynamics. UCRL-JC-139820. 2000.
[42]Peery J S and Carroll D E. Multi-material ALE methods in unstructured grids.Comput. Meth. Appl. Mech. Eng. 2000, 187:591-619.
[43]Margolin L G and Shashkov M. Remapping, recovery and repair on staggered grid. Comput. Meth. Appl. Mech. Eng. 2004, 193:4139-4155.
[44]Richtmyer R D and Morton K W. Difference methods for initial value problems. John Wiley. 1967.
[45]Dukowicz J K, et al. A general topology method. J. Comput. Phys. 1989, 82: 29-63.
[46]Dukowicz J K and Metlz B. Vorticity errors in multidimensional Lagrangian codes. J. Comput. Phys. 1992, 99:115-134.
[47]Desprs B. and Mazeran C. Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Rational Mech. Anal. 2005,178:327-372.
[48]Maire P H, et al. A cell-centered Lagrangian scheme for two dimensional compressible flow problems. SIAM Journal on Scientific Computing. 2007, 29(4):1781-1824.
[49]李文绚,王家浚,钱为义.用于二维非定常气体力学问题的一个拉格朗日差分格式及网格的重新划分.第一届计算流体力学会议论文选集. 1982,237-245.
[50]李德元等.关于拉氏坐标系中二维流体力学有限体积法的精确度.计算物理实验室年报. 1993, 160-165.
[51]Hui W H, Li P Y and Li Z W. A unified coordinate system for solving the two-dimensional Euler equations. J. Comput. Phys. 1999, 153:596-637.
[52]汤华中,邬华谟.高分辨KFVS有限体积方法及其CFD应用.计算数学.1999, 21 (3):375-384.
[53]吴坚,梁珍璇,李荫藩.拉格朗日坐标下的三角形网格重分技术.数值计算与计算机应用. 1994, 4:261-272.
[54]李荫藩,梁珍璇,谈庆明.数值模拟二维高速碰撞问题的拉格朗日任意三角形网格有限体积法.计算物理. 1995, 12(4):433-441.
[55]梁珍璇,李荫藩.轴对称高速碰撞问题的拉格朗日无结构网格有限体积法的并行计算.爆炸与冲击. 1997, 17(3):221-227.
[56]Cheng J. and Shu C W. A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. J. Comput. Phys. 2008, 227:1567-1596.
[57]Wilkins M L. Calculation of elastic-plastic flow. UCRL-7322, Rev.1. Lawrence Livermore Laboratory.
[58]Grandey R. Application of finite difference methods to problems in two- dimensional hydrodynamics. Aeronutronic Publication No. U-1118. California. 1961.
[59]Winslow A M. Equipotential zoning of two dimensional meshes. UCRL-7312. Lawrence Livermore National Laboratory. 1963.
[60]Winslow A M. Equipotential zoning of the interior of a three-dimensional mesh. Report to LSTC. 1990.
[61]Budge K G. ALE shock calculations using stabilized serendipity rezoning scheme. 1991. http://citeseer.ist.psu.edu/cache/papers/cs/20527/http:zSzzSzsherpa.sandia.govzSz9231homezSzpdfpaperszSzAPS9106.pdf/budge91ale.pdf.
[62]Knupp P, et al. Reference Jacobian optimization -Based rezone strategies for arbitrary Lagrangian Eulerian methods. J. Comput. Phys. 2002, 176:93-128.
[63]Shashkov M and Knupp P. Optimization-based reference-matrix rezone strategies for arbitrary Lagrangian Eulerian methods on unstructured meshes. Selcuk J. Appl. Math. 2002, 3(1): 81-99.
[64]Vachal P, et al. Untangling of 2D meshes in ALE simulations. J. Comput. Phys. 2004, 196:627-644.
[65] Garimella R V, et al. Optimization of surface mesh quality using local parameterization. in: Proceedings of the Eleventh International Meshing Roundtable, Ithaca, NY, September 2002, Sandia National Laboratories.
[66]勇珩,袁国兴等.适应于流场变化的网格重构方法.计算物理. 2007, 24(5): 506-511.
[67]姚彦忠等.推进方式的参考Jacobian网格优化方法.计算物理. 2007, 24(3): 253-260.
[68]Dukowicz J K. Conservative rezoning (remapping) for general quadrilateral meshes. J. Comput. Phys. 1984, 54:411-423.
[69]Horak H G, et al. An algorithm for the discrete rezoning of Lagrangian meshes. J. Comput. Phys. 1978, 26:277-284.
[70]Ramshaw J D. Conservative rezoning algorithm for generalized two- dimensional meshes. J. Comput. Phys.1985, 59:193-199.
[71]Ramshaw J D. Simplified second-order rezoning algorithm for generalized two-dimensional meshes. J. Comput. Phys. 1986, 67:214-222.
[72]Dukowicz J K and Kodis J W. Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM J Sci. Stat. Comput. 1987, 8(3):305-321.
[73]Van Leer B. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys. 1977, 23:276-299.
[74]Benson D J. Momentum advection on a staggered mesh. J. Comput. Phys. 1992, 100:143-162.
[75]Margolin L G and Shashkov M. Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys. 2003, 184:266 -298.
[76]Kucharik M, Shashkov M and Wendroff B. An efficient linearity and bound preserving remapping methods. J. Comput. Phys. 2003, 188:462--471.
[77]Loubere R and Shashkov M J. A subcell remapping method on staggered polygonal grids for arbitrary Lagrangian Eulerian Methods. J. Comput. Phys. 2005, 209: 105-138.
[78]Kucharik M and Shashkov M J. Extension of efficient, swept-intergration- based conservative remapping method for meshes with changing connectivity. Int. J. Numer. Meth. in Fluids. 2008,56(8):1359-1365.
[79]Zhang B L, Chen J and Liu X P. A new algorithm used in rezone techniques.计算物理实验室年报.1994, 227-341.
[80]蔡庆东.非结构网格和结构网格生成及自适应软件的研制[博士后报告].北京:北京应用物理与计算数学研究所计算物理实验室. 1999.
[81]王瑞利,毛志明.任意网格重映的样条逼近算法.计算物理. 2001, 18(5): 429 -434.
[82]王瑞利.散乱物理量逼近的插值重映算法.计算物理. 2005, 22(4):299-305.
[83]喻虹.质点积分守恒重映方法.爆炸与冲击. 2003, 23(6):493-500.
[84]Cheng J and Shu C W. A high order accurate conservative remapping method on staggered meshes. Applied Numerical Mathematics.2007.
[85]符尚武,戴自换,邬吉明.二维Lagrange网格的积分守恒重映方法.计算物理. 2005, 22(2):184-188.
[86]温万治等.多边形交错网格的守恒重映算法.计算物理. 2006, 23(5):511-517.
[87]Harten A. Preliminary results on the extension of ENO schemes to two- dimensional problems. In Proceedings, International Conference on NonlinearHyperbolic Problems,Saint-Etienne, 1986, Lecture Notes in Mathematics, edited by Carasso C.(Springer-Verlag Berlin,1987).
[88]Casper J. Finite-volume implementation of high-order essentially non oscillatory schemes in two dimensions. AIAA Journal. 1992, 30(12): 2829 -2835.
[89]Casper J and Atkins H L. A finite volume high-order ENO scheme for two- dimensional hyperbolic systems. J. Comput. Phys. 1993, 106:62-76.
[90]Qiu J X, Shu C W. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 2005, 26(3):907-929.
[91]Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations,edited by A. Quarteroni, Editor, Lecture Notes inMathematics, CIME subseries (Springer-Verlag, Berlin/New York);ICASE Report 97-65.
[92]Abgrall R. Essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation. J. Comput. Phys. 1994, 114:45-58.
[93]刘儒勋,舒其望.计算流体力学的若干新方法.北京:科学出版社. 2003.
[94]Hu C Q and Shu C W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1999, 150:97-127.
[95]Friedrich O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys. 1998, 144:194-212.
[96]Shi J, Hu C Q and Shu C W. A technique of treating negative weights in WENO schemes. J. Comput. Phys. 2002, 175:108-127.
[97]Qiu J X, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 2003, 193:115-135.
[98]Shu C W. High order ENO and WENO schemes for computational fluid dynamics. in High-Order Methods for Computational Physics. T.J. Barth and H. Deconinck, editors, Lecture Notes in Computational Science and Engineering. Springer. 1999, 9:439-582.
[99]Hu C and Shu C W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1990, 150:97-127.
[100]Liu X D, Osher S and Chan T. Weighted essentially Non-oscillatory schemes. J. Comput. Phys. 1994, 115:202-228.
[101]李荫藩,宋松和.一种简化的三阶精度加权ENO格式.数值计算与计算机应用. 1998, 2:127-134.
[102]朱君.流体力学数值方法及并行策略研究[博士学位论文].南京:南京航空航天大学. 2006.
[103]Winslow A. Numerical solution of the quasi-linear Possion equation. J. Comput. Phys. 1967, 1:149-172.
[104]Brackbill J U. An adaptive grid with directional control. J. Comput. Phys. 1993, 108:38-50.
[105]Brackbill J U and Saltzman J S. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 1982, 46:342-368.
[106]Ren W Q and Wang X P. An iterative grid redistribution method for singular problems in multiple dimensions. J. Comput. Phys. 2000, 159:246-273.
[107]Wang D S and Wang X P. Three dimensional adaptive method based on iterative grid redistribution. J. Comput. Phys. 2004, 199:423-436.
[108]Miller K and Miller R N. Moving finite element I. SIAM J. Numer. Anal. 1981, 18:1019-1032.
[109]Davis S F and Flaherty J E. An adaptive finite element method for initial boundary value problems for partial differential equations. SIAM J. Sci. Stat. Comput. 1982, 3:6-27.
[110]Cao W M, Huang W Z and Russell R D. An r-adaptive finite element method based upon moving mesh PDEs. J. Comput. Phys. 1999, 149:221-244.
[111]Li S and Petzold L. Moving mesh methods with upwinding schemes for time- dependent PDEs. J. Comput. Phys. 1997, 131:368-377.
[112]Ceniceros H D and Hou T Y. An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 2001, 172:609-639.
[113]Dvinsky A S. Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 1991, 95:450-476.
[114]Di Y N, et al. Moving mesh finite element methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 2005, 26:1036-1056.
[115]Li R, Tang T and Zhang P W. Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 2001, 170:562-588.
[116]Li R, Tang T and Zhang P W. A moving mesh finite element algorithm for singular problems in two and three space dimensions. J. Comput. Phys. 2002, 177:365-393.
[117]Harten A and Hyman J M. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 1983, 50:235-269.
[118]Tang H Z and Tang T. Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws. SIAM J. Numen. Anal. 2003(41): 487-515.
[119]李德元,徐国荣等.二维非定常流体力学数值方法.北京:科学出版社. 1987.
[120]王春武.高精度ENO格式的构造及并行计算研究[博士学位论文].南京:南京航空航天大学. 2000.
[121]Berger M J and Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 1984, 53:484-512.
[122]Berger M J and Colella P. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 1989, 82(1):64-84.
[123]Anderson R, et al. An arbitrary Lagrangian-Eulerian method with local adaptive mesh refinement for modeling shock hydrodynamics. AIAA 2002-0738.
[124]王瑞利等.数值求解中网格自适应加密和合并技术的研究.数值计算与计算机应用. 2004, 2:145-154.
[125]Morrell J M, et al. A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations. J. Comput. Phys. 2007, 226: 1152-1180.
[126]Shu C W. Essentially non-escillatory and weighted essentially non- oscillatory schemes for hyperbolic conservation laws. in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. B. Cockburn, C. Johnson, C.W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics. Springer. Berlin. 1998, 1697:325-432.
[127]Shu C W, et al. High-order ENO schemes applied to two- and three- dimensional compressible flow. Applied Numerical Mathematics. 1992, 9:45-71.
[128]Ceniceros H D and Hou T Y. An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 2001, 172:609-639.
[129]Sod G A. A survey of several finite difference methods for systems ofnonlinear hyperbolic conservation laws. J. Comput. Phys. 1978, 27:1-31.
[130]Sedov L I. Similarity and dimensional methods in mechanics. Academic Press. New York. 1959.
[131]朱自强.应用计算流体力学.北京:北京航空航天大学出版社.1998.
[132]Peery J S and Carroll D E. Multimaterial ALE methods in unstructured grids. Comput. Meth. Appl. Mech. Eng. 2000, 187:591-619.
[133]Noh W F. Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys. 1987, 72:78-120.
[134]Shu C W and Osher S. Efficient implementation of essentially non- oscillatory shock capturing schemes II. J. Comput. Phys. 1989, 83:32-78.
[135]Michael F. Multidimensional upwinding Part I. The method of transport for solving the Euler equations. J. Comput. Phys. 1998, 143: 159-180.
[136]Colella Ph. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 1990, 87:171-200.
[137]Woodward P R and Colella Ph. The numerical simulation of two dimensional fluid flow with strong shocks. J. Comput. Phys. 1984, 54: 115-173.
[139]赵宁,余彦,唐维军. R-M不稳定性数值模拟方法.计算数学. 2001, 23(4): 477 -490.
[140]唐维军,赵宁.随机扰动下三维流体界面不稳定性的并行计算.计算物理. 2001, 18 (6):539-543.
[2]Von Neumann J and Richtmyer R D. A method for the numerical calculations of hydrodynamics shocks. J. Appl. Phys. 1950, 21: 232.
[3]Kolsky H G. A method for the numerical solution of transient hydrodynamics shock problem in two space dimensions. Report LA-1867, Los Alamos Sci. Lab. 1955.
[4]Browne P L. REZONE, A proposal for accomplishing rezoning in two- dimensional Lagrangian hydrodynamics problems. Report LA-3455-MS, Los Alamos Sci. Lab. 1966.
[5]Liu W K, Chang H and Belytscko T. Arbitrary Lagrangian-Eulerian Petrov- Galerkin finite elements for nonlinear continue. Comput. Meth. Appl. Mech. Eng. 1988, 68(3):259-310.
[6]Nomura T. ALE finite element computations of fluid-structure interaction problems. Comput. Meth. Appl. Mech. Eng. 1994, 112(1/4):291-308.
[7]Brooks A N and Hughes T R. Streamline upwind Petrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 1982, 32(1-3):199- 259.
[8]Babu S S and Bhattacharyya S K. Finite element analysis of fiuid-structure interaction effect on liquid retaining structures due to sloshing. Computers & Structure. 1996, 59(6):1105-1171.
[9]Liu W K and Chen J S. Adaptive ALE finite element with particular reference to external work rate on frictionalinterface. Comput. Meth. Appl. Mech. Eng. 1991, 93:189-216.
[10]Huerta A and Liu W K. Viscous flow with large free surface motion. Comput. Meth. Appl. Mech. Eng. 1988, 69(3):277-324.
[11]Chorin A J. Numerical solution of the Navier-Stokes equation. Math. Comput. 1968, 22(104):745-762.
[12]Ushijima S. Three dimensional arbitrary Lagrangian-Eulerian numerical prediction methods for nonlinear freesurface oscillation. Int. J. Numer. Meth. in Fluids. 1998, 26:605-623.
[13]Benson D J. Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Meth. Appl. Mech. Eng. 1992, 99:235-394.
[14]Darlington R M, et al. A study of ALE simulations of Rayleigh-Taylor instability. Computer Physics Communications. 2002, 135(1): 58-73.
[15]Peterkin R E, et al. Transport of magnetic flux in an arbitrary coordinate ALE code. J. Comput. Phys. 1998, 140 (1):148-171.
[16]Shestakov A L, et al. "The ICF3D Code". Comp. Meth. Appl. Mech. Eng. 2000, 187: 181-201.
[17]Smith R W. AUSM (ALE): a geometrically conservative arbitrary Langrangian -Eulerian flux splitting scheme. J. Comput. Phys. 1999, 150 (1):268-286.
[18]Arber T D, et al. A staggered grid, Lagrangian- Eulerian remap code for 3-D MHD simulations. J. Comput. Phys. 2001, 171(1):151-181.
[19]Rieben R N, et al. An arbitrary Lagrangian Eulerian discretization of MHD on 3D unstructured grids. J. Comput. Phys. 2007,226(1):534-570.
[20]Pierce T and Rodrigue G. A parallel two-sided contact algorithm in ALE3D. Comput. Methods Appl. Mech. Eng. 2005, 194 (27):3127–3146.
[21]Nichols A, et al. Coupled thermal/chemcial/mechanical modeling of energetic materials in ALE3D. UCRL-JC-124706. Lawrence Livermore National Laboratory. 1996.
[22]Couch R, et al. Application of ALE techniques to metal forming simulations. UCRL-JC-114851. Lawrence Livermore National Laboratory. 1993.
[23]Mcclell R, et al. ALE3D model predictions and materials charcterization for the cookoff response of PBXN-109. UCRL-JC-145756. Lawrence Livermore National Laboratory. 2002.
[24]Couch R and Faux D. Simulation of underwater benchmark experiments with ALE3D. UCRL-JC-123819. Lawrence Livermore National Laboratory. 1997.
[25] Stevens D. Compressible multiphase flows in an ALE framework. UCRL-JC -201373. Lawrence Livermore National Laboratory. 2003.
[26]Aro C, Dube E, Futral W. Coupled mechanical/heat transfer simulation on MPP platforms using a finite element linear solver interface. UCRL-JC- 131746. Lawrence Livermore National Laboratory. 1999.
[27]Tipton R. A 2D lagrange MHD code. UCRL-JC-94277. Lawrence Livermore National Laboratory.
[28]Tipton R. Modeling flux compression generators with a 2D ALE code. UCRL -JC-99900. Lawrence Livermore National Laboratory.
[29]阚明先等.衬套内爆ALE方法二维MHD数值模拟.四川大学学报. 2007, 44 (1): 91-96.
[30]Margolin L G. Introduction to“An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds”. J. Comput. Phys. 1997, 135:198-202.
[31]Goad W B. WAT: A numerical method for two-dimensional unsteady fluid flow. LAMS-2365. Los Alamos Sci. Lab. 1960.
[32]Herrmann W. Comparison of finite difference expressions used in Lagrangian fluid flow calculations. AFWL-TR-64-104. Air Force Weapon Lab. 1964.
[33]Wilkins M L. Calculation of elastic-plastic flow. Methods in Computational Physics. 1964, 3:211.
[34]Schulz W D. Two-dimensional Lagrangian hydrodynamic difference equations. Methods in Computational Physics. 1964, 3:1.
[35]Maenchen G and Sack S. The tensor code. Methods in Computational Physics. 1964, 3:181.
[36]Browne P L and Wallick K B. The reduction of mesh tangling in two- dimensional Lagrangian hydrodynamics codes by the use of viscosity, artificial viscosity and TTS, for long, thin zones. LA-4740-MS. Los Alamos Sci. Lab. 1971.
[37]Amsden A A, et al. SALE-3D:A simplified ALE computer program for fluid flow at all flow speed. Los Alamos Scientic Laboratory. 1980.
[38]Amsden A A, et al. SALE-3D: A simplified ALE computer program for calculating three-dimensional fluid flow. Los Alamos Scientic Laboratory. 1981.
[39]Benson D J. An efficient, accurate, simple ALE method for nonlinear finite element programs. Comput. Meth. Appl. Mech. Eng. 1989, 72:305-350.
[40]Benson D J. Vectorization techniques for explicit arbitrary Lagrangian- Eulerian calculations. Comput. Meth. Appl. Mech. Eng. 1992, 96:303-328.
[41]Pember R, et al. A comparison of staggered-mesh Lagrange plus remap and cell-centered direct Eulerian Godunov schemes for Eulerian shock hydrodynamics. UCRL-JC-139820. 2000.
[42]Peery J S and Carroll D E. Multi-material ALE methods in unstructured grids.Comput. Meth. Appl. Mech. Eng. 2000, 187:591-619.
[43]Margolin L G and Shashkov M. Remapping, recovery and repair on staggered grid. Comput. Meth. Appl. Mech. Eng. 2004, 193:4139-4155.
[44]Richtmyer R D and Morton K W. Difference methods for initial value problems. John Wiley. 1967.
[45]Dukowicz J K, et al. A general topology method. J. Comput. Phys. 1989, 82: 29-63.
[46]Dukowicz J K and Metlz B. Vorticity errors in multidimensional Lagrangian codes. J. Comput. Phys. 1992, 99:115-134.
[47]Desprs B. and Mazeran C. Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Rational Mech. Anal. 2005,178:327-372.
[48]Maire P H, et al. A cell-centered Lagrangian scheme for two dimensional compressible flow problems. SIAM Journal on Scientific Computing. 2007, 29(4):1781-1824.
[49]李文绚,王家浚,钱为义.用于二维非定常气体力学问题的一个拉格朗日差分格式及网格的重新划分.第一届计算流体力学会议论文选集. 1982,237-245.
[50]李德元等.关于拉氏坐标系中二维流体力学有限体积法的精确度.计算物理实验室年报. 1993, 160-165.
[51]Hui W H, Li P Y and Li Z W. A unified coordinate system for solving the two-dimensional Euler equations. J. Comput. Phys. 1999, 153:596-637.
[52]汤华中,邬华谟.高分辨KFVS有限体积方法及其CFD应用.计算数学.1999, 21 (3):375-384.
[53]吴坚,梁珍璇,李荫藩.拉格朗日坐标下的三角形网格重分技术.数值计算与计算机应用. 1994, 4:261-272.
[54]李荫藩,梁珍璇,谈庆明.数值模拟二维高速碰撞问题的拉格朗日任意三角形网格有限体积法.计算物理. 1995, 12(4):433-441.
[55]梁珍璇,李荫藩.轴对称高速碰撞问题的拉格朗日无结构网格有限体积法的并行计算.爆炸与冲击. 1997, 17(3):221-227.
[56]Cheng J. and Shu C W. A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. J. Comput. Phys. 2008, 227:1567-1596.
[57]Wilkins M L. Calculation of elastic-plastic flow. UCRL-7322, Rev.1. Lawrence Livermore Laboratory.
[58]Grandey R. Application of finite difference methods to problems in two- dimensional hydrodynamics. Aeronutronic Publication No. U-1118. California. 1961.
[59]Winslow A M. Equipotential zoning of two dimensional meshes. UCRL-7312. Lawrence Livermore National Laboratory. 1963.
[60]Winslow A M. Equipotential zoning of the interior of a three-dimensional mesh. Report to LSTC. 1990.
[61]Budge K G. ALE shock calculations using stabilized serendipity rezoning scheme. 1991. http://citeseer.ist.psu.edu/cache/papers/cs/20527/http:zSzzSzsherpa.sandia.govzSz9231homezSzpdfpaperszSzAPS9106.pdf/budge91ale.pdf.
[62]Knupp P, et al. Reference Jacobian optimization -Based rezone strategies for arbitrary Lagrangian Eulerian methods. J. Comput. Phys. 2002, 176:93-128.
[63]Shashkov M and Knupp P. Optimization-based reference-matrix rezone strategies for arbitrary Lagrangian Eulerian methods on unstructured meshes. Selcuk J. Appl. Math. 2002, 3(1): 81-99.
[64]Vachal P, et al. Untangling of 2D meshes in ALE simulations. J. Comput. Phys. 2004, 196:627-644.
[65] Garimella R V, et al. Optimization of surface mesh quality using local parameterization. in: Proceedings of the Eleventh International Meshing Roundtable, Ithaca, NY, September 2002, Sandia National Laboratories.
[66]勇珩,袁国兴等.适应于流场变化的网格重构方法.计算物理. 2007, 24(5): 506-511.
[67]姚彦忠等.推进方式的参考Jacobian网格优化方法.计算物理. 2007, 24(3): 253-260.
[68]Dukowicz J K. Conservative rezoning (remapping) for general quadrilateral meshes. J. Comput. Phys. 1984, 54:411-423.
[69]Horak H G, et al. An algorithm for the discrete rezoning of Lagrangian meshes. J. Comput. Phys. 1978, 26:277-284.
[70]Ramshaw J D. Conservative rezoning algorithm for generalized two- dimensional meshes. J. Comput. Phys.1985, 59:193-199.
[71]Ramshaw J D. Simplified second-order rezoning algorithm for generalized two-dimensional meshes. J. Comput. Phys. 1986, 67:214-222.
[72]Dukowicz J K and Kodis J W. Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM J Sci. Stat. Comput. 1987, 8(3):305-321.
[73]Van Leer B. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys. 1977, 23:276-299.
[74]Benson D J. Momentum advection on a staggered mesh. J. Comput. Phys. 1992, 100:143-162.
[75]Margolin L G and Shashkov M. Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys. 2003, 184:266 -298.
[76]Kucharik M, Shashkov M and Wendroff B. An efficient linearity and bound preserving remapping methods. J. Comput. Phys. 2003, 188:462--471.
[77]Loubere R and Shashkov M J. A subcell remapping method on staggered polygonal grids for arbitrary Lagrangian Eulerian Methods. J. Comput. Phys. 2005, 209: 105-138.
[78]Kucharik M and Shashkov M J. Extension of efficient, swept-intergration- based conservative remapping method for meshes with changing connectivity. Int. J. Numer. Meth. in Fluids. 2008,56(8):1359-1365.
[79]Zhang B L, Chen J and Liu X P. A new algorithm used in rezone techniques.计算物理实验室年报.1994, 227-341.
[80]蔡庆东.非结构网格和结构网格生成及自适应软件的研制[博士后报告].北京:北京应用物理与计算数学研究所计算物理实验室. 1999.
[81]王瑞利,毛志明.任意网格重映的样条逼近算法.计算物理. 2001, 18(5): 429 -434.
[82]王瑞利.散乱物理量逼近的插值重映算法.计算物理. 2005, 22(4):299-305.
[83]喻虹.质点积分守恒重映方法.爆炸与冲击. 2003, 23(6):493-500.
[84]Cheng J and Shu C W. A high order accurate conservative remapping method on staggered meshes. Applied Numerical Mathematics.2007.
[85]符尚武,戴自换,邬吉明.二维Lagrange网格的积分守恒重映方法.计算物理. 2005, 22(2):184-188.
[86]温万治等.多边形交错网格的守恒重映算法.计算物理. 2006, 23(5):511-517.
[87]Harten A. Preliminary results on the extension of ENO schemes to two- dimensional problems. In Proceedings, International Conference on NonlinearHyperbolic Problems,Saint-Etienne, 1986, Lecture Notes in Mathematics, edited by Carasso C.(Springer-Verlag Berlin,1987).
[88]Casper J. Finite-volume implementation of high-order essentially non oscillatory schemes in two dimensions. AIAA Journal. 1992, 30(12): 2829 -2835.
[89]Casper J and Atkins H L. A finite volume high-order ENO scheme for two- dimensional hyperbolic systems. J. Comput. Phys. 1993, 106:62-76.
[90]Qiu J X, Shu C W. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 2005, 26(3):907-929.
[91]Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations,edited by A. Quarteroni, Editor, Lecture Notes inMathematics, CIME subseries (Springer-Verlag, Berlin/New York);ICASE Report 97-65.
[92]Abgrall R. Essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation. J. Comput. Phys. 1994, 114:45-58.
[93]刘儒勋,舒其望.计算流体力学的若干新方法.北京:科学出版社. 2003.
[94]Hu C Q and Shu C W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1999, 150:97-127.
[95]Friedrich O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys. 1998, 144:194-212.
[96]Shi J, Hu C Q and Shu C W. A technique of treating negative weights in WENO schemes. J. Comput. Phys. 2002, 175:108-127.
[97]Qiu J X, Shu C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 2003, 193:115-135.
[98]Shu C W. High order ENO and WENO schemes for computational fluid dynamics. in High-Order Methods for Computational Physics. T.J. Barth and H. Deconinck, editors, Lecture Notes in Computational Science and Engineering. Springer. 1999, 9:439-582.
[99]Hu C and Shu C W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1990, 150:97-127.
[100]Liu X D, Osher S and Chan T. Weighted essentially Non-oscillatory schemes. J. Comput. Phys. 1994, 115:202-228.
[101]李荫藩,宋松和.一种简化的三阶精度加权ENO格式.数值计算与计算机应用. 1998, 2:127-134.
[102]朱君.流体力学数值方法及并行策略研究[博士学位论文].南京:南京航空航天大学. 2006.
[103]Winslow A. Numerical solution of the quasi-linear Possion equation. J. Comput. Phys. 1967, 1:149-172.
[104]Brackbill J U. An adaptive grid with directional control. J. Comput. Phys. 1993, 108:38-50.
[105]Brackbill J U and Saltzman J S. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 1982, 46:342-368.
[106]Ren W Q and Wang X P. An iterative grid redistribution method for singular problems in multiple dimensions. J. Comput. Phys. 2000, 159:246-273.
[107]Wang D S and Wang X P. Three dimensional adaptive method based on iterative grid redistribution. J. Comput. Phys. 2004, 199:423-436.
[108]Miller K and Miller R N. Moving finite element I. SIAM J. Numer. Anal. 1981, 18:1019-1032.
[109]Davis S F and Flaherty J E. An adaptive finite element method for initial boundary value problems for partial differential equations. SIAM J. Sci. Stat. Comput. 1982, 3:6-27.
[110]Cao W M, Huang W Z and Russell R D. An r-adaptive finite element method based upon moving mesh PDEs. J. Comput. Phys. 1999, 149:221-244.
[111]Li S and Petzold L. Moving mesh methods with upwinding schemes for time- dependent PDEs. J. Comput. Phys. 1997, 131:368-377.
[112]Ceniceros H D and Hou T Y. An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 2001, 172:609-639.
[113]Dvinsky A S. Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 1991, 95:450-476.
[114]Di Y N, et al. Moving mesh finite element methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 2005, 26:1036-1056.
[115]Li R, Tang T and Zhang P W. Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 2001, 170:562-588.
[116]Li R, Tang T and Zhang P W. A moving mesh finite element algorithm for singular problems in two and three space dimensions. J. Comput. Phys. 2002, 177:365-393.
[117]Harten A and Hyman J M. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 1983, 50:235-269.
[118]Tang H Z and Tang T. Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws. SIAM J. Numen. Anal. 2003(41): 487-515.
[119]李德元,徐国荣等.二维非定常流体力学数值方法.北京:科学出版社. 1987.
[120]王春武.高精度ENO格式的构造及并行计算研究[博士学位论文].南京:南京航空航天大学. 2000.
[121]Berger M J and Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 1984, 53:484-512.
[122]Berger M J and Colella P. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 1989, 82(1):64-84.
[123]Anderson R, et al. An arbitrary Lagrangian-Eulerian method with local adaptive mesh refinement for modeling shock hydrodynamics. AIAA 2002-0738.
[124]王瑞利等.数值求解中网格自适应加密和合并技术的研究.数值计算与计算机应用. 2004, 2:145-154.
[125]Morrell J M, et al. A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations. J. Comput. Phys. 2007, 226: 1152-1180.
[126]Shu C W. Essentially non-escillatory and weighted essentially non- oscillatory schemes for hyperbolic conservation laws. in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. B. Cockburn, C. Johnson, C.W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics. Springer. Berlin. 1998, 1697:325-432.
[127]Shu C W, et al. High-order ENO schemes applied to two- and three- dimensional compressible flow. Applied Numerical Mathematics. 1992, 9:45-71.
[128]Ceniceros H D and Hou T Y. An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 2001, 172:609-639.
[129]Sod G A. A survey of several finite difference methods for systems ofnonlinear hyperbolic conservation laws. J. Comput. Phys. 1978, 27:1-31.
[130]Sedov L I. Similarity and dimensional methods in mechanics. Academic Press. New York. 1959.
[131]朱自强.应用计算流体力学.北京:北京航空航天大学出版社.1998.
[132]Peery J S and Carroll D E. Multimaterial ALE methods in unstructured grids. Comput. Meth. Appl. Mech. Eng. 2000, 187:591-619.
[133]Noh W F. Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys. 1987, 72:78-120.
[134]Shu C W and Osher S. Efficient implementation of essentially non- oscillatory shock capturing schemes II. J. Comput. Phys. 1989, 83:32-78.
[135]Michael F. Multidimensional upwinding Part I. The method of transport for solving the Euler equations. J. Comput. Phys. 1998, 143: 159-180.
[136]Colella Ph. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 1990, 87:171-200.
[137]Woodward P R and Colella Ph. The numerical simulation of two dimensional fluid flow with strong shocks. J. Comput. Phys. 1984, 54: 115-173.
[139]赵宁,余彦,唐维军. R-M不稳定性数值模拟方法.计算数学. 2001, 23(4): 477 -490.
[140]唐维军,赵宁.随机扰动下三维流体界面不稳定性的并行计算.计算物理. 2001, 18 (6):539-543.