浅水方程高分辨率算法的研究
|
摘要
本文以近似Riemann解为基础,建立了一、二维带源项浅水方程的高精度、高分辨率守恒型模型。应用特征分解和迎风处理源项的方法,保证守恒型计算格式的“和谐性”。通过改进的底坡计算方法及守恒型通量限制方式,较好地解决了干底Riemann解处理动边界和总质量守恒等问题,确保了计算基本无质量误差。建立了耦合的二维守恒型浅水水质模型,采用修正浓度的方法,使浓度计算结果合理,保证水体和污染物浓度同时保持数值质量守恒。本文主要工作如下：
1.以Roe格式的近似Riemann解为基础,应用通量限制法和MUSCL型坡度限制法分别建立了求解一维非平底浅水流动方程的高分辨率Godunov格式,同时采用改进的两步Runge-Kutta方法获得时间二阶精度。为了使方程左边的对流项与方程右边的底坡源项始终保持“和谐”,对源项采用迎风特征分解,以满足守恒性。讨论了使用不同通量限制器时,对流量和水位计算结果的影响。
2.逐点插入法可以对任意给定复杂边界的二维区域按照需求进行加密网格,本文设计了一种基于Bowyer-Watson逐点插入算法的Delaunay网格生成方法,为二维浅水方程计算提供了有力的保证。运用不需添加辅助点的边界恢复方法,在初始网格生成时就满足Delaunay网格准则。在加点的次序和选择加点位置上使用了一种比较合理的机制,使得生成的加密网格比较规则。在加密点的插入问题上,通过三角形网络的拓扑关系,利用两点是否位于一条直线的异侧和拓扑关系来解决这一问题,提高了新点插入的效率,使生成加密网格的时间大大缩短。
3.提出了一种处理带有干湿界面的基本无质量误差的高分辨率守恒型数值格式。采用非结构化网格的有限体积法,对底坡源项采用特征分解保证“和谐性”,摩擦力源项采用半隐式格式增加格式的稳定性。通过改进底坡的计算方法及守恒型通量限制方式,解决了计算过程中干湿界面(如露滩问题)处理不当引起的非静水解、出现负水深单元、质量不守恒等诸多问题。在此基础上,构造了一种基于MUSCL格式的PLCD (Project Limited Central Difference)方法,通过对底坡源项进行相应的数值修正以满足淹没情况下的静水问题,保证高分辨率格式的“和谐性”,使格式的时空均达到二阶精度。
4.基于本文构造的MUSCL格式的新型PLCD方法,建立了耦合的高分辨率二维守恒型浅水水质模型。采用修正浓度的方法,使浓度计算结果合理且满足守恒性。考虑到对流扩散方程的稳定性由对流和扩散共同决定、时间步长由Peclet数和CFL数共同决定等问题,讨论了不同扩散系数影响下的显式和隐式求解扩散项技术的差异。由于隐式计算扩散项方法为无条件稳定,时间步长完全由对流项决定,能够有效避免时间步长过小的问题,同时具有较小的耗散性。数值实验表明该模型计算的浓度解具有较好的稳定性和光滑性,保证水体和污染物浓度同时保持数值质量守恒,能够较好地模拟水流水质问题。
5.进一步讨论了MUSCL格式的几种梯度算子(LCD, Durlofsky, PLCD, MLG, MLG-Wierse)在水流水质问题数值计算中的性能,并且通过一些经典算例将一阶及其它二阶格式进行了比较分析。数值实验表明PLCD格式、MLG格式和MLG-Wierse格式三种格式计算结果最佳,且新构造的PLCD格式在计算量少于MLG和MLG-Wierse两种格式,可以作为一种高精度、高分辨率的格式应用于水流水质问题的实际模拟。
Based on the finite volume method and Roe's approximate Riemann solver, the One-dimensional and Two-dimensional conservative high-resolution shallow water models with source terms are developed in this study.The discretization of the bed slope source terms is done following an upwind approach to protect the scheme harmonious. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model ensures a global conservation and positive values of both water level and solute concentration. The main work of this article is as following:
1. A technique has been investigated for extending Roe's finite volume method to second-order spatial and temporal accurate MUSCL-type slope-limiting approach in order to simulate shallow water flow over uneven beds. The discretization of the bed slope source terms is done following an upwind approach to ensure convective term from left side of the equation and bottom slope source term from right side of the equation to preserve the "humanity" of conservative numerical scheme. The objective of this study is to compare the performances corresponding to different variables of reconstruction to determine whether there exists all optimal approach.
2. A Delaunay mesh generation method based on Bowyer-Watson method is designed for a complex region. Based on the initial mesh, a edge recovery algorithm by no adding Steiner points is presented. A new technique considering the order and the location of insertion points is established to generate high quality mesh. Many methods are adopted to improve the quality and efficiency of the mesh.
3. A wetting/drying condition (WDC) for unsteady shallow water flow leading to zero mass error is presented. The WDC has been incorporated into a cell-centred finite volume method based on Roe's approximate Riemann solver on unstructured grids. The discretization of the bed slope source terms was done following all upwind approach and the semi-implicit treatment was used for the friction source terms. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. A new PLCD method based on two-dimensional MUSCL-type finite volume schemes is developed. Numerical results are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real complex geometries and bottom slope variation.
4. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model corrects solute concentration to keep the results reasonable. This paper discusses the difference of diffusion terms between explicit and implicit discretization. The centered discretization of the diffusion terms is in an implicit way in this model. Numerical experiments show that this model has good stability and smoothness and ensures conservation of the quality of water and solute concentration. Therefore, it could be used for the simulation of solute transport.
5. A PLCD (Project Limited Central Difference) technique has been investigated based on MUSCL-type slope-limiting approach. The performances corresponding to different variables of reconstruction (LCD, Durlofsky, PLCD, MLG and MLG-Wierse) are compared. Numerical results indicate that PLCD, MLG and MLG-Wierse are efficient and the new PLCD computation expense of the new constructed PLCD scheme is much less than MLG and MLG-Wierse.
1.以Roe格式的近似Riemann解为基础,应用通量限制法和MUSCL型坡度限制法分别建立了求解一维非平底浅水流动方程的高分辨率Godunov格式,同时采用改进的两步Runge-Kutta方法获得时间二阶精度。为了使方程左边的对流项与方程右边的底坡源项始终保持“和谐”,对源项采用迎风特征分解,以满足守恒性。讨论了使用不同通量限制器时,对流量和水位计算结果的影响。
2.逐点插入法可以对任意给定复杂边界的二维区域按照需求进行加密网格,本文设计了一种基于Bowyer-Watson逐点插入算法的Delaunay网格生成方法,为二维浅水方程计算提供了有力的保证。运用不需添加辅助点的边界恢复方法,在初始网格生成时就满足Delaunay网格准则。在加点的次序和选择加点位置上使用了一种比较合理的机制,使得生成的加密网格比较规则。在加密点的插入问题上,通过三角形网络的拓扑关系,利用两点是否位于一条直线的异侧和拓扑关系来解决这一问题,提高了新点插入的效率,使生成加密网格的时间大大缩短。
3.提出了一种处理带有干湿界面的基本无质量误差的高分辨率守恒型数值格式。采用非结构化网格的有限体积法,对底坡源项采用特征分解保证“和谐性”,摩擦力源项采用半隐式格式增加格式的稳定性。通过改进底坡的计算方法及守恒型通量限制方式,解决了计算过程中干湿界面(如露滩问题)处理不当引起的非静水解、出现负水深单元、质量不守恒等诸多问题。在此基础上,构造了一种基于MUSCL格式的PLCD (Project Limited Central Difference)方法,通过对底坡源项进行相应的数值修正以满足淹没情况下的静水问题,保证高分辨率格式的“和谐性”,使格式的时空均达到二阶精度。
4.基于本文构造的MUSCL格式的新型PLCD方法,建立了耦合的高分辨率二维守恒型浅水水质模型。采用修正浓度的方法,使浓度计算结果合理且满足守恒性。考虑到对流扩散方程的稳定性由对流和扩散共同决定、时间步长由Peclet数和CFL数共同决定等问题,讨论了不同扩散系数影响下的显式和隐式求解扩散项技术的差异。由于隐式计算扩散项方法为无条件稳定,时间步长完全由对流项决定,能够有效避免时间步长过小的问题,同时具有较小的耗散性。数值实验表明该模型计算的浓度解具有较好的稳定性和光滑性,保证水体和污染物浓度同时保持数值质量守恒,能够较好地模拟水流水质问题。
5.进一步讨论了MUSCL格式的几种梯度算子(LCD, Durlofsky, PLCD, MLG, MLG-Wierse)在水流水质问题数值计算中的性能,并且通过一些经典算例将一阶及其它二阶格式进行了比较分析。数值实验表明PLCD格式、MLG格式和MLG-Wierse格式三种格式计算结果最佳,且新构造的PLCD格式在计算量少于MLG和MLG-Wierse两种格式,可以作为一种高精度、高分辨率的格式应用于水流水质问题的实际模拟。
Based on the finite volume method and Roe's approximate Riemann solver, the One-dimensional and Two-dimensional conservative high-resolution shallow water models with source terms are developed in this study.The discretization of the bed slope source terms is done following an upwind approach to protect the scheme harmonious. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model ensures a global conservation and positive values of both water level and solute concentration. The main work of this article is as following:
1. A technique has been investigated for extending Roe's finite volume method to second-order spatial and temporal accurate MUSCL-type slope-limiting approach in order to simulate shallow water flow over uneven beds. The discretization of the bed slope source terms is done following an upwind approach to ensure convective term from left side of the equation and bottom slope source term from right side of the equation to preserve the "humanity" of conservative numerical scheme. The objective of this study is to compare the performances corresponding to different variables of reconstruction to determine whether there exists all optimal approach.
2. A Delaunay mesh generation method based on Bowyer-Watson method is designed for a complex region. Based on the initial mesh, a edge recovery algorithm by no adding Steiner points is presented. A new technique considering the order and the location of insertion points is established to generate high quality mesh. Many methods are adopted to improve the quality and efficiency of the mesh.
3. A wetting/drying condition (WDC) for unsteady shallow water flow leading to zero mass error is presented. The WDC has been incorporated into a cell-centred finite volume method based on Roe's approximate Riemann solver on unstructured grids. The discretization of the bed slope source terms was done following all upwind approach and the semi-implicit treatment was used for the friction source terms. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. A new PLCD method based on two-dimensional MUSCL-type finite volume schemes is developed. Numerical results are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real complex geometries and bottom slope variation.
4. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model corrects solute concentration to keep the results reasonable. This paper discusses the difference of diffusion terms between explicit and implicit discretization. The centered discretization of the diffusion terms is in an implicit way in this model. Numerical experiments show that this model has good stability and smoothness and ensures conservation of the quality of water and solute concentration. Therefore, it could be used for the simulation of solute transport.
5. A PLCD (Project Limited Central Difference) technique has been investigated based on MUSCL-type slope-limiting approach. The performances corresponding to different variables of reconstruction (LCD, Durlofsky, PLCD, MLG and MLG-Wierse) are compared. Numerical results indicate that PLCD, MLG and MLG-Wierse are efficient and the new PLCD computation expense of the new constructed PLCD scheme is much less than MLG and MLG-Wierse.
引文
[1]谭维炎.计算浅水动力学-有限体积法的应用[M].北京：清华大学出版社,1998.
[2]McDonald P W. The computation of transonic flow through two-dimensional gas turbine cascades [J]. American Society of Mechanical Engineers,1971,71-89.
[3]Godunov S K. A finite difference method for the numerical computation of discontinuous solution of the fluid dynamics [J]. Mat. Sb.,1959,47:271-290.
[4]Stanley Osher, Fred Solomon. Upwing difference schemes for hyperbolic systems of conservation laws [J]. Mathematics of Computation,1982,38(158):339-374.
[5]Steger J L, Warming R F. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods [J]. Journal of Computational Physics,1981,40: 263-293.
[6]Roe P L. Approximate Riemann solvers, parameter vectors, and difference schemes [J]. J. Comput. Phys.,1981,43(2):357-371.
[7]Van Leer B. Towards the ultimate conservative difference scheme II:Monotonicity and conservation combined in a second order scheme [J]. J. Comput. Phys.,1974,14:361-370.
[8]水鸿寿.一维流体力学差分方法[M].北京：国防工业出版社,1998.
[9]Toro EF. Riemann solvers and numerial methods for fluid dynamics [M]. Berlin:Springer, 1999.
[10]Lios M S, Steffen C J. A new flux splitting scheme [J]. J. Comput. Phys,1993,107 (23):23-39.
[11]Lios M S. A sequel to AUSM:AUSM+[J]. J. Comput. Phys,1996,129:364-382.
[12]王平,朱自强,吕晓斌.二维非结构自适应多重网格的Euler方程解[J].计算物理,2000,17(5)：497-503.
[13]Casulli V. Semi-implicit finite difference methods for the two-dimensional shallow water equations [J]. Journal of Computational Physics,1990,86:56-74.
[14]Casulli V, Cattani E. Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow [J]. Computers and Application,1994,27(4): 99-112.
[15]Oliveira A, Baptista M. On the role of tracking on Eulerian-Langrangian solutions of the transport equation [J]. Advances in Water Resources,1998,21:539-544.
[16]Ham D. A., Pietrzak J., Stelling G. S. A streamline tracking algorithm for semi-Lagrangian advection schemes based on the analytic integration of the velocity field [C]. Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering,2004:182-190.
[17]Martin N., Gorelick S. M. Semi-analytical method for departure point determination [J]. International Journal for Numerical Methods in Fluids,2005,47:121-137.
[18]Casulli V., Cheng P. Semi-implicit finite difference methods for three-dimensional shallow water flow [J]. International Journal for Numerical Methods in Fluids,1992, 15:629-648.
[19]Leonard B. P. Stability of explicit advection schemes. The balance point location rule [J]. International Journal for Numerical Methods in Fluids,2002,38(5):471-514.
[20]温生辉,蔡启富,汤军健,蔡篙.厦门海域浅水三维潮流常动力模型[J].海洋学报,2003,25(2)：1-17.
[21]Harten A. High resolution schemes for hyperbolic conservation laws [J]. J. Comput. Phys.,1983,49:357-393.
[22]王嘉松,倪汉根,金生等.用TVD显隐格式模拟溃坝波的演进与反射[J].水利学报,1998,5：7-11.
[23]王嘉松,倪汉根,金生.二维溃坝坡传播和绕流特性的高精度数值模拟[J].水利学报,1998,10：1-6.
[24]Goodman J B, Leveque R J. On the accuracy of stable schemes for 2D scalar conservation laws [J]. Math. Comput.,1985,45:15-21.
[25]Harten A, Osher S. Uniformly high order accurate essentially non-oscillatory schemes I [J]. SIAM J. Sci. Comput.,1987,24:279-309.
[26]Liu X D, Osher S, Chan T. Weighted essentially non-oscillatory schemes [J]. J. Comput. Phys.,1994,115:200-212.
[27]Van Leer B. Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method [J]. J. Comput. Phys.1979,32(1):101-136.
[28]Durlofsky L J, Engquist B, Osher S. Triangle based adaptive stencils for the solution of hyperbolic conservation laws [J]. J. Comput. Phys.,1992,98(1):64-73.
[29]Batten P, Lambert C, Causon D M. Positively conservative high-resolution convection schemes for unstructured elements [J]. Int. J. Numer. Methods Eng.,1996,39(11):1821-1838.
[30]Wierse M. A new theoretically motivated higher order upwind scheme on unstructured grids of simplices [J]. Advances in Computational Mathematics,1997,7:303-335.
[31]Hubbard M E. Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids [J]. Journal of Computational Physics,1999,155(1):54-74.
[32]刘儒勋,舒其望.计算流体力学的若千新方法[M].北京：科学出版社,2003.
[33]陶文铨.计算传热学的近代进展[M].北京：科学出版社,2000.
[34]陶文铨.数值传热学(第二版)[M].西安：西安交通大学出版社,2001.
[35]谭维炎,胡四一.浅水流动计算中一阶有限体积法Osher格式的实现[J].水科学进展.1994,5(4)：262-270.
[36]Zhao D H, Shen H W, Tabios III G Q, Lai J S, Tan W Y. Finite-volume two-dimensional unsteady-flow model for river basins [J]. J. Hydraul. Eng.,1994,120(7):863-883.
[37]Zhao D H, Shen H W, Lai J S and Tabios G Q. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling [J]. J. of Hydraul. Eng.,1996,122(12):692-702.
[38]潘存鸿,林炳尧,毛献忠.求解二维浅水流动方程的Godunov格式[J].水动力学研究与进展,A辑,2003,18(1)：16-23.
[39]潘存鸿.浅水间断流动数值模拟及其在钱塘江河口涌潮分析中的应用[D].上海大学博士学位论文,2007.
[40]Bermudez A, Vazquez M E. Upwind methods for hyperbolic conservation laws with source terms [J]. Computers & Fluids,1994,23(8):1049-1071.
[41]Vazquez-Cendon M E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry [J]. J. Comput. Phys.1999,148:497-526.
[42]LeVeque R J. Balancing source terms and flux gradients in high-resolution Godunov methods:The quasisteady wave-propagation algorithm [J]. J. Comput. Phys.,1998, 146:346-365.
[43]Hubbard M E, Garcia-Navarro P. Flux difference splitting and the balancing of source terms and flux gradients [J]. J. Comput. Phys.,2000,165(1):89-125.
[44]Garcia-Navarro P, Vazquez-Cendon M E. On numerical treatment of the source terms in the shallow water equations [J]. Comput. Fluids,2000,29:951-979.
[45]Alcrudo F, Garcia-Navarro P. A high-resolution Godunov-type scheme in finite volume for the 2D shallow-water equations [J]. Int. J. Numer. Methods Fluids,1993,16:489-505.
[46]Brufau P, Garcia-Navarro P. Two-dimensional dam break flow simulation [J]. Int. J. Numer. Meth. Fluids,2000,33:35-57.
[47]Burguete J, Garcia-Navanro P. Efficient construction of high-resolution TVD conservative schemes for equations with source terms:Application to shallow water flows [J]. Int. J. Numer. Meth. Fluids,2001,37:209-248.
[48]Brufau P, Vazquez-Cendon M E, Garcia-Navarro P. A numerical model for the flooding and drying of irregular domains [J]. International Journal for Numerical Methods in Fluids, 2002,39(3):247-275.
[49]Brufau P, Garcia-Navarro P. Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique [J]. Journal of Computational Physics, 2003,186(2):503-526.
[50]Brufau P, Garcia-Navatro P, Vazquez-Cendon M E. Zero mass error using unsteady wetting-drying conditions in shallow flow over dry irregular topography [J]. Int. J. Numer. Meth. Fluids,2004,45(10):1047-1082.
[51]王志力,耿艳芬,金生.具有复杂计算域和地形的二维浅水流动数值模拟,水利学报,2005,36(4)：1-9.
[52]Burguete J, Garcia-Navarro P. Efficient construction of high-resolution TVD conservative schemes for equations with source terms:Application to shallow water flows [J].Int. J. Numer. Methods in Fluids,2001,37:209-248.
[53]Zhou J G, Causon D M, Mingham C G, Ingram D M. The surface gradient method for the treatment of source terms in the shallow-water equations [J]. J. Comput. Phys.,2001, 168(1):1-25.
[54]Zhou J G, et al. Numerical solutions of the the shallow water equations with discontinuous bed topography [J]. Int. J. Numer. Meth. Fluids,2002,38:769-788.
[55]Zhou J G, Causon D M, Mingham C G, et al. Numerical predication of dam-break flows in general geometries with complex bed topography [J]. J Hydraul Eng,2004,130(4):332-340.
[56]于普兵.二维浅水水流数值模拟技术研究-无结构网格有限体积法[D],南京水利科学研究院硕士学位论文,2006.
[57]王立辉.溃坝水流数值模拟与溃坝风险分析研究[D],南京水利科学研究院博士学位论文,2006.
[58]Xing Y, Shu C W. High order finite difference WENO schemes with the exact conservation property for the shallow water equations [J]. Journal of Computational Physics,2005, 208:206-227.
[59]Mohammadian A, Le Roux D Y. Simulation of shallow flows over variable topographies using unstructured grids [J]. Int. J. Numer. Meth. Fluids,2006,52:473-498.
[60]何少荃,王连祥.窄缝法在二维边界变动水域计算中的应用[J].水利学报,1986,(12)：11-19.
[61]程文辉,王船海.用正交曲线网格及“冻结法”计算河道流速场[J].水利学报,1988,(6)：16-25.
[62]彭凯,方铎,曹叔尤.在二维流动计算中应用“河床切削”技术处理边界问题[J].水动力学研究与进展A辑,1992,7(2)：200-205.
[63]魏文礼,金忠青.复杂边界河道流速场的数值模拟[J].水利学报,1994,11：26-30.
[64]胡四一,谭维炎.无结构网格上二维浅水流动的数值模拟[J].水科学进展,1995,6(1)：1-9.
[65]潘存鸿,林炳尧,毛献忠.浅水问题的动边界数值模拟[J].水利水运工程学报,2004,4：1-7.
[66]朱德军,陈永灿,刘昭伟.处理二维浅水流动中动边界问题的淹没节点法[J].水动力学研究与进展A辑,200621(1)：102-106.
[67]孙健,陶建华.潮流数值模拟中动边界处理方法研究[J].水动力学研究与进展A辑,2007,22(1)：44-52.
[68]Kawahara M, Umetsu T. Finite element method for moving boundary problems in river flow [J]. International Journal for Numerical Methods in Fluids,1986,6(6):365-386.
[69]Meselhe E A, Holly F M. Simulation of unsteady flow in irrigation canals with a dry bed [J]. Journal of Hydraulic Engineering,1993,119(9):1021-1030.
[70]Tchamen G W, Kahawita R. The numerical simulation of wetting and drying areas using Riemann solvers[C]. Proceedings of Modeling of Flood Propagation over Initially Dry Areas, Milan, Italy,1994.
[71]Eleuterio F Toro. Shock-capturing methods for free-surface shallow flows [M]. Wiley New York,2001.
[72]吴时强,丁道扬.剖开算子法解具有自由表面的平面紊流速度场[J].水利水运科学研究,1992,1：39-48.
[73]吴时强,丁道扬,吴碧君,马英.平面二维动态水质数学模型[J].水动力学研究与进展A辑,1996,11(6)：653-660.
[74]华祖林.弯曲河段水流水质二维数值模拟[J].水资源保护,1999,3：12-15.
[75]赵棣华,戚晨,庚维德,徐葆华,裴中平.平面二维水流-水质有限体积法及黎曼近似解模型[J].水科学进展,2000 11(4)：368-374.
[76]赵棣华,姚琪,蒋艳,杨珏,逢勇.通量向量分裂格式的二维水流-水质模拟[J].水科学进展,2002,13(6)：701-706.
[77]赵棣华,沈福新,颜志俊,卢恭和.基于有限体积法及IDS格式的感潮河段二维泥沙冲淤模型[J].水动力学研究与进展A辑,2004,19(1)：98-103.
[78]龚春生,姚琪,赵棣华等.浅水湖泊平面二维水流水质底泥污染模型研究[J].水科学进展,2006,17(4)：496-501.
[79]Nakamura T, Tanaka R, Yabe T, Takizawa K. Exactly conservative semi-Lagrangian scheme for multidimensional hyperbolic equations with directional splitting technique [J]. Journal of Computational Physics,2001,174:171-207.
[80]Burguete J, Garcia-Navarro P. Semi-Lagrangian conservative schemes versus Eulerian schemes to solve advection in river flow transport [C]. Proceedings of the Fifth International Conference on Hydroinformatics, Cardiff, U.K.,2002,181-185.
[81]Begnudelli L, Sanders F. Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying [J]. Journal of Hydraulic Engineering, 2006,132(4):371-384.
[82]Nujic M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows [J]. J. Hydraul. Res.,1995,33(1):101-111.
[83]Bradford S F, Katopodes N D. Finite volume model for nonlevel basin irrigation [J]. Irrig. Drain,2001,127(4):216-223.
[84]Bradford S F, Sanders B F. Finite-volume model for shallow-water flooding of arbitrary topography [J]. J. Hydraul. Eng.,2002,128(3):289-298.
[85]Sleigh P A, Gaskell P K. An unstructured finite-volume algorithm for predicting flow in rivers and estuaries [J]. Computers & Fluids,1998,27(4):479-508.
[86]Christoph Schar, Smolarkiewicz P K. A synchronous and iterative flux-correction formalism for coupled transport equations [J]. J. Comput. Phys.,1996,128(1):101-120.
[87]耿艳芬,王志力,金生.一维浅水方程的高精度GODUNOV格式[J].水动力学研究与进展A辑,2005,20(4)：507-512.
[88]王志力,耿艳芬,金生.带源项浅水方程的通量平衡离散[J].水科学进展,2005,16(3)：373-379.
[89]Voronoi G. Nouvelles applications des parameters continus a la theorie des formes quadratiques, deuxieme memorie:Recherches sur les parrallelloedres primitives [J]. Jounal fur die Reine und Angewandte Mathematik,1908,134:198-287.
[90]Delaunay B. Sur la sphere vide [J]. Bulletin of the Academy of Sciences of the USSR, Classe des Sciences Mathematiques et Naturelles,1934 (8):793-800.
[91]Bowyer A. Computing Dirichlet Tessellations [J]. The Computer Jounal,1981,24: 162-166.
[92]Watson D. Computing the N-dimensional Delaunay Tessellation with application to Voronoy polytopes [J]. The Computer Journal,1981,24:167-172.
[93]Hermeline F. Une methode automatique de mailage en dimension [D]. Paris:Universite, 1980.
[94]Lawson C L. Software for C1 surface interpolation in mathematical software III [M]. New York:Academic Press,1977:161-194.
[95]Tsai V J D. Delaunay triangulations in TIN creation:an overview and a linear-time algorithm [J]. Int. J. of GIS,1993,7(6):501-524.
[96]Shamos M I, Hoey D. Closest-point problems [C]. Proceedings of the 16th Annual Symposium on the Foundations of Computer Science,1975:151-162.
[97]Lewis B A, Robinson J S. Triangulation of planar regions with appli cat ions [J]. The Computer Journal,1978,21(4):324-332.
[98]Lee D T, Schachter B J. Two algorithms for constructing a Delaunay triangulation [J]. Int. J. of Computer and Information Sciences,1980,9(3):219-242.
[99]Green P J and Sibson R. Computing Dirichlet Tesselations in the plane [J]. The Computer Journal,1978,21 (2):168-173.
[100]Brassel K E, Reif D. Procedure to generate Thiessen polygons [J]. Geophysical Analysis, 1979,11:289-303.
[101]McCaullaghM T, Ross C G. Delaunay triangulation of a random data set for iirarithmic mapping [J]. The Cartographic Journal,1980,17:93-99.
[102]Maus A. Delaunay triangulation and the convex hull of n points in expected linear time [J]. BIT,1984,24:151-163.
[103]Sloan S W. A fast algorithm for constructing Delaunay triangulations in the plane [J]. Andvanced Engineering Software,1987,9:34-55.
[104]Macedonio G, Pareschi M T. An algorithm for the triangulation of arbitrarily distributed points:Applications to volume estimate and terrain fitting [J]. Computers & Geosciences,1991,17:859-874.
[105]陈建军.非结构化网格生成及其并行化的若干问题研究[D].浙江：浙江大学,2006.
[106]Zheng Y, Lewis R W, Gethin D T. Three-dimensional unstructured mesh generation:Part 1. Fundamental aspects of triangulation and point creation [J]. Computer Methods in Applied Mechanics and Engineering,1996,134:249-268.
[107]Weatherill N P, Hassan 0. Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints [J]. International Journal for Numerical Methods in Engineering,1994,37:2005-2039.
[108]George P L, Hecht F, Saltel E. Automatic mesh generator with specified boundary [J]. Computer Methods in Applied Mechanics and Engineering,1991,92:269-288.
[109]Rebay S. Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer-Watson algorithm [J]. Journal of Computational Physics,1993,106:125-138.
[110]吴宇晓,张登荣.生成Delaunay三角网的快速合成算法[J].浙江大学学报(理学版),2004,5：343-348.
[111]刘少华,程朋根,陈红华,等.Delaunay三角网内插特征点算法研究[J].华东地质学院学报,2002,9：254-257.
[112]蒲浩,宋占峰,詹振炎.快速构建三角网数字地形模型方法的研究[J].中国铁道科学,2002,(12)：100-105.
[113]徐道柱,刘海砚.Delaunay三角网建立的改进算法[J].测绘与空间地理信息,2007,30(1)：38-41.
[114]Ming-Hseng Tseng, CHIAR Chu. Two-dimensional shallow water flows simulation using TVD-MacCormack scheme [J]. J Hydraul Res,2000,38(2):123-131.
[115]Zoppou C, Roberts S. Numerical solution of the two-dimensional unsteady dam break [J]. Applied Mathematical Modeling,2000,24:457-475.
[116]Macchione F, Morelli M A. Pratical aspects in comparing shock-capturing schemes for dam break problems [J]. Journal of Hydraulic Engineering (ASCE),2002,129(3):187-195.
[117]Thacker W C. Some exact solutions to the nonlinear shallow-water wave equations [J]. J. Fluid Mech.,1981,107:499-500.
[118]Murillo J, Garcfa-Navarro P, Burguete J, Brufau P. A conservative 2D model of inundation flow with solute transport over dry bed [J]. International Journal for Numerical Methods in Fluids,2006,52(10):1059-1092.
[119]Murillo J, Burguete J, Brufau P, Garcia-Navarro P. Coupling between shallow water and solute flow equations:analysis and management of source terms in 2D [J]. International Journal for Numerical Methods in Fluids,2005,49(3):267-299.
[120]Murillo J, Garcia-Navarro P, Burguete J. Analysis of a second-order upwind method for the simulation of solute transport in 2D shallow water flow [J]. International Journal for Numerical Methods in Fluids,2008,56(6):661-686.
[121]汪继文,刘儒勋.间断解问题的有限体积法[J].计算物理,2001,18(2)：98-105.
[122]Van Leer B. Towards the ultimate conservative difference scheme III. Upstream-centered finite difference schemes for ideal compressible flow [J]. J. Comput. Phys.,1977(23):263-275.
[123]Anastasiou K, Chan C T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes [J]. Inter. J. for Numerical Methods in Fluids,1997,24:1225-1245.
[124]Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws [R]. Lecture Notes in Mathematics,1998,1697:325-432.
[125]Liu X D. A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws [J]. SIAM J. Numer. Anal., 1993,30(3):701-716.
[126]Tae Hoon Yoon, Seok-Koo Kang. Finite volume model for two-dimensional shallow water flows on unstructured grids [J]. J.Hydraul Eng.,2004,130:678-688.
[127]Hu H, Mingham C G, Causon D M. A bore-capturing finite volume method for open-channel flows [J]. Int. J. Numer. Methods in Fluids,1998,28:1241-1261.
[128]Roger B, Fui jhara M, Borthwick A G L. Adaptive Q-tree godunov-type scheme for shallow water equations [J]. Int. J. Numer. Methods in Fluids,2001,35:247-280.
[129]王立辉,胡四一,龚春生.二维浅水方程的非结构网格数值解.水利水运工程学报,2006,1：8-13.
[130]Tamamidis P, Assanis D N. Evaluation of various high-order-accuracy schemes with and without flux limiters [J]. Int. J. Numer. Methods Fluids,1993,16:931-948.
[2]McDonald P W. The computation of transonic flow through two-dimensional gas turbine cascades [J]. American Society of Mechanical Engineers,1971,71-89.
[3]Godunov S K. A finite difference method for the numerical computation of discontinuous solution of the fluid dynamics [J]. Mat. Sb.,1959,47:271-290.
[4]Stanley Osher, Fred Solomon. Upwing difference schemes for hyperbolic systems of conservation laws [J]. Mathematics of Computation,1982,38(158):339-374.
[5]Steger J L, Warming R F. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods [J]. Journal of Computational Physics,1981,40: 263-293.
[6]Roe P L. Approximate Riemann solvers, parameter vectors, and difference schemes [J]. J. Comput. Phys.,1981,43(2):357-371.
[7]Van Leer B. Towards the ultimate conservative difference scheme II:Monotonicity and conservation combined in a second order scheme [J]. J. Comput. Phys.,1974,14:361-370.
[8]水鸿寿.一维流体力学差分方法[M].北京：国防工业出版社,1998.
[9]Toro EF. Riemann solvers and numerial methods for fluid dynamics [M]. Berlin:Springer, 1999.
[10]Lios M S, Steffen C J. A new flux splitting scheme [J]. J. Comput. Phys,1993,107 (23):23-39.
[11]Lios M S. A sequel to AUSM:AUSM+[J]. J. Comput. Phys,1996,129:364-382.
[12]王平,朱自强,吕晓斌.二维非结构自适应多重网格的Euler方程解[J].计算物理,2000,17(5)：497-503.
[13]Casulli V. Semi-implicit finite difference methods for the two-dimensional shallow water equations [J]. Journal of Computational Physics,1990,86:56-74.
[14]Casulli V, Cattani E. Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow [J]. Computers and Application,1994,27(4): 99-112.
[15]Oliveira A, Baptista M. On the role of tracking on Eulerian-Langrangian solutions of the transport equation [J]. Advances in Water Resources,1998,21:539-544.
[16]Ham D. A., Pietrzak J., Stelling G. S. A streamline tracking algorithm for semi-Lagrangian advection schemes based on the analytic integration of the velocity field [C]. Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering,2004:182-190.
[17]Martin N., Gorelick S. M. Semi-analytical method for departure point determination [J]. International Journal for Numerical Methods in Fluids,2005,47:121-137.
[18]Casulli V., Cheng P. Semi-implicit finite difference methods for three-dimensional shallow water flow [J]. International Journal for Numerical Methods in Fluids,1992, 15:629-648.
[19]Leonard B. P. Stability of explicit advection schemes. The balance point location rule [J]. International Journal for Numerical Methods in Fluids,2002,38(5):471-514.
[20]温生辉,蔡启富,汤军健,蔡篙.厦门海域浅水三维潮流常动力模型[J].海洋学报,2003,25(2)：1-17.
[21]Harten A. High resolution schemes for hyperbolic conservation laws [J]. J. Comput. Phys.,1983,49:357-393.
[22]王嘉松,倪汉根,金生等.用TVD显隐格式模拟溃坝波的演进与反射[J].水利学报,1998,5：7-11.
[23]王嘉松,倪汉根,金生.二维溃坝坡传播和绕流特性的高精度数值模拟[J].水利学报,1998,10：1-6.
[24]Goodman J B, Leveque R J. On the accuracy of stable schemes for 2D scalar conservation laws [J]. Math. Comput.,1985,45:15-21.
[25]Harten A, Osher S. Uniformly high order accurate essentially non-oscillatory schemes I [J]. SIAM J. Sci. Comput.,1987,24:279-309.
[26]Liu X D, Osher S, Chan T. Weighted essentially non-oscillatory schemes [J]. J. Comput. Phys.,1994,115:200-212.
[27]Van Leer B. Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method [J]. J. Comput. Phys.1979,32(1):101-136.
[28]Durlofsky L J, Engquist B, Osher S. Triangle based adaptive stencils for the solution of hyperbolic conservation laws [J]. J. Comput. Phys.,1992,98(1):64-73.
[29]Batten P, Lambert C, Causon D M. Positively conservative high-resolution convection schemes for unstructured elements [J]. Int. J. Numer. Methods Eng.,1996,39(11):1821-1838.
[30]Wierse M. A new theoretically motivated higher order upwind scheme on unstructured grids of simplices [J]. Advances in Computational Mathematics,1997,7:303-335.
[31]Hubbard M E. Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids [J]. Journal of Computational Physics,1999,155(1):54-74.
[32]刘儒勋,舒其望.计算流体力学的若千新方法[M].北京：科学出版社,2003.
[33]陶文铨.计算传热学的近代进展[M].北京：科学出版社,2000.
[34]陶文铨.数值传热学(第二版)[M].西安：西安交通大学出版社,2001.
[35]谭维炎,胡四一.浅水流动计算中一阶有限体积法Osher格式的实现[J].水科学进展.1994,5(4)：262-270.
[36]Zhao D H, Shen H W, Tabios III G Q, Lai J S, Tan W Y. Finite-volume two-dimensional unsteady-flow model for river basins [J]. J. Hydraul. Eng.,1994,120(7):863-883.
[37]Zhao D H, Shen H W, Lai J S and Tabios G Q. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling [J]. J. of Hydraul. Eng.,1996,122(12):692-702.
[38]潘存鸿,林炳尧,毛献忠.求解二维浅水流动方程的Godunov格式[J].水动力学研究与进展,A辑,2003,18(1)：16-23.
[39]潘存鸿.浅水间断流动数值模拟及其在钱塘江河口涌潮分析中的应用[D].上海大学博士学位论文,2007.
[40]Bermudez A, Vazquez M E. Upwind methods for hyperbolic conservation laws with source terms [J]. Computers & Fluids,1994,23(8):1049-1071.
[41]Vazquez-Cendon M E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry [J]. J. Comput. Phys.1999,148:497-526.
[42]LeVeque R J. Balancing source terms and flux gradients in high-resolution Godunov methods:The quasisteady wave-propagation algorithm [J]. J. Comput. Phys.,1998, 146:346-365.
[43]Hubbard M E, Garcia-Navarro P. Flux difference splitting and the balancing of source terms and flux gradients [J]. J. Comput. Phys.,2000,165(1):89-125.
[44]Garcia-Navarro P, Vazquez-Cendon M E. On numerical treatment of the source terms in the shallow water equations [J]. Comput. Fluids,2000,29:951-979.
[45]Alcrudo F, Garcia-Navarro P. A high-resolution Godunov-type scheme in finite volume for the 2D shallow-water equations [J]. Int. J. Numer. Methods Fluids,1993,16:489-505.
[46]Brufau P, Garcia-Navarro P. Two-dimensional dam break flow simulation [J]. Int. J. Numer. Meth. Fluids,2000,33:35-57.
[47]Burguete J, Garcia-Navanro P. Efficient construction of high-resolution TVD conservative schemes for equations with source terms:Application to shallow water flows [J]. Int. J. Numer. Meth. Fluids,2001,37:209-248.
[48]Brufau P, Vazquez-Cendon M E, Garcia-Navarro P. A numerical model for the flooding and drying of irregular domains [J]. International Journal for Numerical Methods in Fluids, 2002,39(3):247-275.
[49]Brufau P, Garcia-Navarro P. Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique [J]. Journal of Computational Physics, 2003,186(2):503-526.
[50]Brufau P, Garcia-Navatro P, Vazquez-Cendon M E. Zero mass error using unsteady wetting-drying conditions in shallow flow over dry irregular topography [J]. Int. J. Numer. Meth. Fluids,2004,45(10):1047-1082.
[51]王志力,耿艳芬,金生.具有复杂计算域和地形的二维浅水流动数值模拟,水利学报,2005,36(4)：1-9.
[52]Burguete J, Garcia-Navarro P. Efficient construction of high-resolution TVD conservative schemes for equations with source terms:Application to shallow water flows [J].Int. J. Numer. Methods in Fluids,2001,37:209-248.
[53]Zhou J G, Causon D M, Mingham C G, Ingram D M. The surface gradient method for the treatment of source terms in the shallow-water equations [J]. J. Comput. Phys.,2001, 168(1):1-25.
[54]Zhou J G, et al. Numerical solutions of the the shallow water equations with discontinuous bed topography [J]. Int. J. Numer. Meth. Fluids,2002,38:769-788.
[55]Zhou J G, Causon D M, Mingham C G, et al. Numerical predication of dam-break flows in general geometries with complex bed topography [J]. J Hydraul Eng,2004,130(4):332-340.
[56]于普兵.二维浅水水流数值模拟技术研究-无结构网格有限体积法[D],南京水利科学研究院硕士学位论文,2006.
[57]王立辉.溃坝水流数值模拟与溃坝风险分析研究[D],南京水利科学研究院博士学位论文,2006.
[58]Xing Y, Shu C W. High order finite difference WENO schemes with the exact conservation property for the shallow water equations [J]. Journal of Computational Physics,2005, 208:206-227.
[59]Mohammadian A, Le Roux D Y. Simulation of shallow flows over variable topographies using unstructured grids [J]. Int. J. Numer. Meth. Fluids,2006,52:473-498.
[60]何少荃,王连祥.窄缝法在二维边界变动水域计算中的应用[J].水利学报,1986,(12)：11-19.
[61]程文辉,王船海.用正交曲线网格及“冻结法”计算河道流速场[J].水利学报,1988,(6)：16-25.
[62]彭凯,方铎,曹叔尤.在二维流动计算中应用“河床切削”技术处理边界问题[J].水动力学研究与进展A辑,1992,7(2)：200-205.
[63]魏文礼,金忠青.复杂边界河道流速场的数值模拟[J].水利学报,1994,11：26-30.
[64]胡四一,谭维炎.无结构网格上二维浅水流动的数值模拟[J].水科学进展,1995,6(1)：1-9.
[65]潘存鸿,林炳尧,毛献忠.浅水问题的动边界数值模拟[J].水利水运工程学报,2004,4：1-7.
[66]朱德军,陈永灿,刘昭伟.处理二维浅水流动中动边界问题的淹没节点法[J].水动力学研究与进展A辑,200621(1)：102-106.
[67]孙健,陶建华.潮流数值模拟中动边界处理方法研究[J].水动力学研究与进展A辑,2007,22(1)：44-52.
[68]Kawahara M, Umetsu T. Finite element method for moving boundary problems in river flow [J]. International Journal for Numerical Methods in Fluids,1986,6(6):365-386.
[69]Meselhe E A, Holly F M. Simulation of unsteady flow in irrigation canals with a dry bed [J]. Journal of Hydraulic Engineering,1993,119(9):1021-1030.
[70]Tchamen G W, Kahawita R. The numerical simulation of wetting and drying areas using Riemann solvers[C]. Proceedings of Modeling of Flood Propagation over Initially Dry Areas, Milan, Italy,1994.
[71]Eleuterio F Toro. Shock-capturing methods for free-surface shallow flows [M]. Wiley New York,2001.
[72]吴时强,丁道扬.剖开算子法解具有自由表面的平面紊流速度场[J].水利水运科学研究,1992,1：39-48.
[73]吴时强,丁道扬,吴碧君,马英.平面二维动态水质数学模型[J].水动力学研究与进展A辑,1996,11(6)：653-660.
[74]华祖林.弯曲河段水流水质二维数值模拟[J].水资源保护,1999,3：12-15.
[75]赵棣华,戚晨,庚维德,徐葆华,裴中平.平面二维水流-水质有限体积法及黎曼近似解模型[J].水科学进展,2000 11(4)：368-374.
[76]赵棣华,姚琪,蒋艳,杨珏,逢勇.通量向量分裂格式的二维水流-水质模拟[J].水科学进展,2002,13(6)：701-706.
[77]赵棣华,沈福新,颜志俊,卢恭和.基于有限体积法及IDS格式的感潮河段二维泥沙冲淤模型[J].水动力学研究与进展A辑,2004,19(1)：98-103.
[78]龚春生,姚琪,赵棣华等.浅水湖泊平面二维水流水质底泥污染模型研究[J].水科学进展,2006,17(4)：496-501.
[79]Nakamura T, Tanaka R, Yabe T, Takizawa K. Exactly conservative semi-Lagrangian scheme for multidimensional hyperbolic equations with directional splitting technique [J]. Journal of Computational Physics,2001,174:171-207.
[80]Burguete J, Garcia-Navarro P. Semi-Lagrangian conservative schemes versus Eulerian schemes to solve advection in river flow transport [C]. Proceedings of the Fifth International Conference on Hydroinformatics, Cardiff, U.K.,2002,181-185.
[81]Begnudelli L, Sanders F. Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying [J]. Journal of Hydraulic Engineering, 2006,132(4):371-384.
[82]Nujic M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows [J]. J. Hydraul. Res.,1995,33(1):101-111.
[83]Bradford S F, Katopodes N D. Finite volume model for nonlevel basin irrigation [J]. Irrig. Drain,2001,127(4):216-223.
[84]Bradford S F, Sanders B F. Finite-volume model for shallow-water flooding of arbitrary topography [J]. J. Hydraul. Eng.,2002,128(3):289-298.
[85]Sleigh P A, Gaskell P K. An unstructured finite-volume algorithm for predicting flow in rivers and estuaries [J]. Computers & Fluids,1998,27(4):479-508.
[86]Christoph Schar, Smolarkiewicz P K. A synchronous and iterative flux-correction formalism for coupled transport equations [J]. J. Comput. Phys.,1996,128(1):101-120.
[87]耿艳芬,王志力,金生.一维浅水方程的高精度GODUNOV格式[J].水动力学研究与进展A辑,2005,20(4)：507-512.
[88]王志力,耿艳芬,金生.带源项浅水方程的通量平衡离散[J].水科学进展,2005,16(3)：373-379.
[89]Voronoi G. Nouvelles applications des parameters continus a la theorie des formes quadratiques, deuxieme memorie:Recherches sur les parrallelloedres primitives [J]. Jounal fur die Reine und Angewandte Mathematik,1908,134:198-287.
[90]Delaunay B. Sur la sphere vide [J]. Bulletin of the Academy of Sciences of the USSR, Classe des Sciences Mathematiques et Naturelles,1934 (8):793-800.
[91]Bowyer A. Computing Dirichlet Tessellations [J]. The Computer Jounal,1981,24: 162-166.
[92]Watson D. Computing the N-dimensional Delaunay Tessellation with application to Voronoy polytopes [J]. The Computer Journal,1981,24:167-172.
[93]Hermeline F. Une methode automatique de mailage en dimension [D]. Paris:Universite, 1980.
[94]Lawson C L. Software for C1 surface interpolation in mathematical software III [M]. New York:Academic Press,1977:161-194.
[95]Tsai V J D. Delaunay triangulations in TIN creation:an overview and a linear-time algorithm [J]. Int. J. of GIS,1993,7(6):501-524.
[96]Shamos M I, Hoey D. Closest-point problems [C]. Proceedings of the 16th Annual Symposium on the Foundations of Computer Science,1975:151-162.
[97]Lewis B A, Robinson J S. Triangulation of planar regions with appli cat ions [J]. The Computer Journal,1978,21(4):324-332.
[98]Lee D T, Schachter B J. Two algorithms for constructing a Delaunay triangulation [J]. Int. J. of Computer and Information Sciences,1980,9(3):219-242.
[99]Green P J and Sibson R. Computing Dirichlet Tesselations in the plane [J]. The Computer Journal,1978,21 (2):168-173.
[100]Brassel K E, Reif D. Procedure to generate Thiessen polygons [J]. Geophysical Analysis, 1979,11:289-303.
[101]McCaullaghM T, Ross C G. Delaunay triangulation of a random data set for iirarithmic mapping [J]. The Cartographic Journal,1980,17:93-99.
[102]Maus A. Delaunay triangulation and the convex hull of n points in expected linear time [J]. BIT,1984,24:151-163.
[103]Sloan S W. A fast algorithm for constructing Delaunay triangulations in the plane [J]. Andvanced Engineering Software,1987,9:34-55.
[104]Macedonio G, Pareschi M T. An algorithm for the triangulation of arbitrarily distributed points:Applications to volume estimate and terrain fitting [J]. Computers & Geosciences,1991,17:859-874.
[105]陈建军.非结构化网格生成及其并行化的若干问题研究[D].浙江：浙江大学,2006.
[106]Zheng Y, Lewis R W, Gethin D T. Three-dimensional unstructured mesh generation:Part 1. Fundamental aspects of triangulation and point creation [J]. Computer Methods in Applied Mechanics and Engineering,1996,134:249-268.
[107]Weatherill N P, Hassan 0. Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints [J]. International Journal for Numerical Methods in Engineering,1994,37:2005-2039.
[108]George P L, Hecht F, Saltel E. Automatic mesh generator with specified boundary [J]. Computer Methods in Applied Mechanics and Engineering,1991,92:269-288.
[109]Rebay S. Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer-Watson algorithm [J]. Journal of Computational Physics,1993,106:125-138.
[110]吴宇晓,张登荣.生成Delaunay三角网的快速合成算法[J].浙江大学学报(理学版),2004,5：343-348.
[111]刘少华,程朋根,陈红华,等.Delaunay三角网内插特征点算法研究[J].华东地质学院学报,2002,9：254-257.
[112]蒲浩,宋占峰,詹振炎.快速构建三角网数字地形模型方法的研究[J].中国铁道科学,2002,(12)：100-105.
[113]徐道柱,刘海砚.Delaunay三角网建立的改进算法[J].测绘与空间地理信息,2007,30(1)：38-41.
[114]Ming-Hseng Tseng, CHIAR Chu. Two-dimensional shallow water flows simulation using TVD-MacCormack scheme [J]. J Hydraul Res,2000,38(2):123-131.
[115]Zoppou C, Roberts S. Numerical solution of the two-dimensional unsteady dam break [J]. Applied Mathematical Modeling,2000,24:457-475.
[116]Macchione F, Morelli M A. Pratical aspects in comparing shock-capturing schemes for dam break problems [J]. Journal of Hydraulic Engineering (ASCE),2002,129(3):187-195.
[117]Thacker W C. Some exact solutions to the nonlinear shallow-water wave equations [J]. J. Fluid Mech.,1981,107:499-500.
[118]Murillo J, Garcfa-Navarro P, Burguete J, Brufau P. A conservative 2D model of inundation flow with solute transport over dry bed [J]. International Journal for Numerical Methods in Fluids,2006,52(10):1059-1092.
[119]Murillo J, Burguete J, Brufau P, Garcia-Navarro P. Coupling between shallow water and solute flow equations:analysis and management of source terms in 2D [J]. International Journal for Numerical Methods in Fluids,2005,49(3):267-299.
[120]Murillo J, Garcia-Navarro P, Burguete J. Analysis of a second-order upwind method for the simulation of solute transport in 2D shallow water flow [J]. International Journal for Numerical Methods in Fluids,2008,56(6):661-686.
[121]汪继文,刘儒勋.间断解问题的有限体积法[J].计算物理,2001,18(2)：98-105.
[122]Van Leer B. Towards the ultimate conservative difference scheme III. Upstream-centered finite difference schemes for ideal compressible flow [J]. J. Comput. Phys.,1977(23):263-275.
[123]Anastasiou K, Chan C T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes [J]. Inter. J. for Numerical Methods in Fluids,1997,24:1225-1245.
[124]Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws [R]. Lecture Notes in Mathematics,1998,1697:325-432.
[125]Liu X D. A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws [J]. SIAM J. Numer. Anal., 1993,30(3):701-716.
[126]Tae Hoon Yoon, Seok-Koo Kang. Finite volume model for two-dimensional shallow water flows on unstructured grids [J]. J.Hydraul Eng.,2004,130:678-688.
[127]Hu H, Mingham C G, Causon D M. A bore-capturing finite volume method for open-channel flows [J]. Int. J. Numer. Methods in Fluids,1998,28:1241-1261.
[128]Roger B, Fui jhara M, Borthwick A G L. Adaptive Q-tree godunov-type scheme for shallow water equations [J]. Int. J. Numer. Methods in Fluids,2001,35:247-280.
[129]王立辉,胡四一,龚春生.二维浅水方程的非结构网格数值解.水利水运工程学报,2006,1：8-13.
[130]Tamamidis P, Assanis D N. Evaluation of various high-order-accuracy schemes with and without flux limiters [J]. Int. J. Numer. Methods Fluids,1993,16:931-948.