基于Roe格式的不规则地形上浅水模拟
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摘要
为准确求解不规则地形上的浅水方程,建立了一种基于Roe格式的黎曼近似解的具有二阶时间空间精度的和谐离散格式.守恒变量采用加入坡度限制的MUSCL(monotonic upstream-centred scheme for conservation laws)格式进行重构.为保证格式和谐性,引入水面梯度法并基于静水条件构造了底坡项的离散格式:河床梯度采用中心差分格式,而水深则为计算单元附近4个重构水深的平均值.将上述重新调整过的Roe-MUSCL格式运用于3个不规则地形上的水流算例,计算结果均比较理想.
A well-balanced numerical scheme based on Roe's approximate Riemann solver is established to solve the shallow water equations on irregular topographies.In the scheme,the predictor-corrector type finite volume scheme is used and the conservative variables are constructed with the MUSCL scheme(Monotonic Upstream-Centred Scheme for Conservation Laws).Thus the scheme is with second-order accuracy both in time and space.To establish the well-balanced scheme,the surface gradient method is introduced and a new discretization scheme of the bed slope source term has been constructed,in which the water depth is calculated by four reconstructed water depths.The well-balanced scheme has been tested by three examples of flow over irregular bed bottoms and satisfactory results have been acquired.
A well-balanced numerical scheme based on Roe's approximate Riemann solver is established to solve the shallow water equations on irregular topographies.In the scheme,the predictor-corrector type finite volume scheme is used and the conservative variables are constructed with the MUSCL scheme(Monotonic Upstream-Centred Scheme for Conservation Laws).Thus the scheme is with second-order accuracy both in time and space.To establish the well-balanced scheme,the surface gradient method is introduced and a new discretization scheme of the bed slope source term has been constructed,in which the water depth is calculated by four reconstructed water depths.The well-balanced scheme has been tested by three examples of flow over irregular bed bottoms and satisfactory results have been acquired.
引文
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[9]朱德军,陈永灿,刘昭伟,等.含底坡项浅水方程的一种数值求解方法[J].水科学进展,2012,23(6):796-801.Zhu Dejun,Chen Yongcan,Liu Zhaowei,et al.Numerical approximation of the shallow water equations with bed slope source term[J].Advances in Water Science,2012,23(6):796-801.
[10]Rogers B D,Borthwick A G L,Taylor P H.Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver[J].Journal of Computational Physics,2003,192(2):422-451.
[11]Capart H,Eldho T I,Huang S Y,et al.Treatment of natural geometry in finite volume river flow computations[J].Journal of Hydraulic Engineering,2003,129(5):385-393.
[12]刘刚,金生.基于修正Roe格式的有限体积法求解二维浅水方程[J].水利水运工程学报,2009,(3):29-33.Liu Gang,Jin Sheng.Finite volume model for the 2D shallow water equations using modified Roe scheme[J].Hydo-Science and Engineering,2009,(3):29-33.
[13]王党伟,陈建国,吉祖稳.不规则地形上浅水模拟平衡性的实现[J].计算力学学报,2012,29(4):604-608.Wang Dangwei,Chen Jianguo,Ji Zuwen.Method for keeping balance in shallow-water simulation over irregular topography[J].Chinese Journal of Computational Mechanics,2012,29(4):604-608.
[14]柏禄海,金生.带源项浅水方程的高阶格式研究[J].水动力学研究与进展,2009,24(1):22-28.Bai Luhai,Jin Sheng.Study on high resolution scheme for shallow water equation with source terms[J].Chinese Journal of Hydrodynamics,2009,24(1):22-28.
[15]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.Wang Zhili,Geng Yanfen,Jin Sheng.Flux balance method for shallow water equation with source terms[J].Advances in Water Science,2005,16(3):373-379.
[16]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[J].水利学报,2005,36(4):439-444.Wang Zhili,Geng Yanfen,Jin Sheng.Numerical modeling of 2Dshallow water clow with complicated geometry and topography[J].Journal of Hydraulic Engineering,2005,36(4):439-444.
[17]Caselles V,Donat R, Haro G.Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes[J].Computers&Fluids,2009,38(1):16-36.
[18]何杰,徐志扬,辛文杰.浅水方程Roe型格式的平衡性[J].河海大学学报(自然科学版),2009,37(4):450-456.He Jie,Xu Zhiyang,Xin Wenjie.Balance of shallow water equations by use of Roe scheme[J].Journal of Hohai University(Natural Sciences),2009,37(4):450-456.
[19]Wang J W,Liu R X.A Comparative study of finite volume methods on unstructured meshes for simulation of 2Dshallow water wave problems[J].Mathematics&Computers in Simulation,2000,53(3):171-184.
[20]夏军强,王光谦,Lin Binliang,等.复杂边界及实际地形上溃坝洪水流动过程模拟[J].水科学进展,2010,21(3):289-298.Xia Junqiang,Wang Guangqian,Lin Binliang,et al.Two-dimensional modelling of dam-break floods over actual terrain with complex geometries using a finite volume method[J]. Advances in Water Science,2010,21(3):289-298.
[21]Toro E F.Riemann Solvers and Numerical Methods for Fluid Dynamics-A Practical Introduction[M].Berlin:Springer,2009:366-400.
[22]Zhou J G,Causon D M,Mingham C G,et al.The surface gradient method for the treatment of source terms in the shallow-water equations[J].Journal of Computational Physics,2001,168(1):1-25.
[23]Ying X,Khan A A,Wang S S Y.Upwind conservative scheme for the saint venant equations[J].Journal of Hydraulic Engineering,2004,130(10):977-987.
[24]Delgado A D,Nieto E F.A family of stable numerical solvers for the shallow water equations with source terms[J].Computer Methods in Applied Mechanics&Engineering,2003,192(2):203-225.
[2] Kesserwani G,Liang Q.Well-balanced RKDG2solutions to the shallow water equations over irregular domains with wetting and drying[J].Computers&Fluids,2010,39:2040-2050.
[3] Hou J,Simons F,Mahgoub M,et al.A robust wellbalanced model on unstructured grids for shallow water flows with wetting and drying over complex topography[J].Computer Methods in Applied Mechanics&Engineering,2013,257(15):126-149.
[4]周浩澜,陈洋波,任启伟.不规则地形浅水模拟[J].水动力学研究与进展,2010,25(5):594-600.Zhou Haolan,Chen Yangbo,Ren Qiwei.The shallow-water simulation for irregular terrain[J].Chinese Journal of Hydrodynamics,2010,25(5):594-600.
[5]王昆,金生,马志强,等.基于和谐性离散格式求解带源项的二维浅水方程[J].水动力学研究与进展,2009,24(5):535-542.Wang Kun,Jin Sheng,Ma Zhiqiang,et al.On a wellbalanced discretization scheme for two-dimensional shallow water equations with source terms[J].Chinese Journal of Hydrodynamics,2009,24(5):535-542.
[6]吕彪,金生,艾丛芳.非结构化网格下求解二维浅水方程的和谐Roe格式[J].水利水运工程学报,2010,(2):39-44.LüBiao,Jin Sheng,Ai Congfang.Well-balanced roetype scheme for 2Dshallow water flow using unstructured grids[J].Hydo-Science and Engineering,2010,(2):39-44.
[7]艾丛芳,金生.基于非结构网格求解二维浅水方程的高精度有限体积方法[J].计算力学学报,2009,26(6):900-905.Ai Congfang,Jin Sheng.Solution of the 2Dshallow water equations using the high-resolution finite-volume method on unstructured meshes[J].Chinese Journal of Computational Mechanics,2009,26(6):900-905.
[8]艾丛芳,金生.求解具有复杂地形二维浅水方程的修正HLL格式[J].大连理工大学学报,2009,49(6):926-931.Ai Congfang,Jin Sheng.Solution of 2Dshallow water equations with complicated geometry using modified HLL scheme[J].Journal of Dalian University of Technology,2009,49(6):926-931.
[9]朱德军,陈永灿,刘昭伟,等.含底坡项浅水方程的一种数值求解方法[J].水科学进展,2012,23(6):796-801.Zhu Dejun,Chen Yongcan,Liu Zhaowei,et al.Numerical approximation of the shallow water equations with bed slope source term[J].Advances in Water Science,2012,23(6):796-801.
[10]Rogers B D,Borthwick A G L,Taylor P H.Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver[J].Journal of Computational Physics,2003,192(2):422-451.
[11]Capart H,Eldho T I,Huang S Y,et al.Treatment of natural geometry in finite volume river flow computations[J].Journal of Hydraulic Engineering,2003,129(5):385-393.
[12]刘刚,金生.基于修正Roe格式的有限体积法求解二维浅水方程[J].水利水运工程学报,2009,(3):29-33.Liu Gang,Jin Sheng.Finite volume model for the 2D shallow water equations using modified Roe scheme[J].Hydo-Science and Engineering,2009,(3):29-33.
[13]王党伟,陈建国,吉祖稳.不规则地形上浅水模拟平衡性的实现[J].计算力学学报,2012,29(4):604-608.Wang Dangwei,Chen Jianguo,Ji Zuwen.Method for keeping balance in shallow-water simulation over irregular topography[J].Chinese Journal of Computational Mechanics,2012,29(4):604-608.
[14]柏禄海,金生.带源项浅水方程的高阶格式研究[J].水动力学研究与进展,2009,24(1):22-28.Bai Luhai,Jin Sheng.Study on high resolution scheme for shallow water equation with source terms[J].Chinese Journal of Hydrodynamics,2009,24(1):22-28.
[15]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.Wang Zhili,Geng Yanfen,Jin Sheng.Flux balance method for shallow water equation with source terms[J].Advances in Water Science,2005,16(3):373-379.
[16]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[J].水利学报,2005,36(4):439-444.Wang Zhili,Geng Yanfen,Jin Sheng.Numerical modeling of 2Dshallow water clow with complicated geometry and topography[J].Journal of Hydraulic Engineering,2005,36(4):439-444.
[17]Caselles V,Donat R, Haro G.Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes[J].Computers&Fluids,2009,38(1):16-36.
[18]何杰,徐志扬,辛文杰.浅水方程Roe型格式的平衡性[J].河海大学学报(自然科学版),2009,37(4):450-456.He Jie,Xu Zhiyang,Xin Wenjie.Balance of shallow water equations by use of Roe scheme[J].Journal of Hohai University(Natural Sciences),2009,37(4):450-456.
[19]Wang J W,Liu R X.A Comparative study of finite volume methods on unstructured meshes for simulation of 2Dshallow water wave problems[J].Mathematics&Computers in Simulation,2000,53(3):171-184.
[20]夏军强,王光谦,Lin Binliang,等.复杂边界及实际地形上溃坝洪水流动过程模拟[J].水科学进展,2010,21(3):289-298.Xia Junqiang,Wang Guangqian,Lin Binliang,et al.Two-dimensional modelling of dam-break floods over actual terrain with complex geometries using a finite volume method[J]. Advances in Water Science,2010,21(3):289-298.
[21]Toro E F.Riemann Solvers and Numerical Methods for Fluid Dynamics-A Practical Introduction[M].Berlin:Springer,2009:366-400.
[22]Zhou J G,Causon D M,Mingham C G,et al.The surface gradient method for the treatment of source terms in the shallow-water equations[J].Journal of Computational Physics,2001,168(1):1-25.
[23]Ying X,Khan A A,Wang S S Y.Upwind conservative scheme for the saint venant equations[J].Journal of Hydraulic Engineering,2004,130(10):977-987.
[24]Delgado A D,Nieto E F.A family of stable numerical solvers for the shallow water equations with source terms[J].Computer Methods in Applied Mechanics&Engineering,2003,192(2):203-225.