一维浅水方程的Runge-Kutta间断有限元数值模拟与应用
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摘要
浅水方程在海洋、河流、气候模拟、海洋生物学和环境等众多领域有着广泛的应用,例如风暴潮、海啸和洪水的预测,泥沙和污染物的输运等。
本文在一维浅水方程组的基础上建立了二阶Runge-Kutta司断有限元数值模型,并成功应用于复杂地形条件下间断浅水流动的数值模拟。本文的主要内容和结论如下：
(1)从守恒形式的浅水方程组出发,对求解区域进行单元剖分,并在单元上对方程进行积分。在模型的空间离散过程中,跨单元边界处采用HLL和HLLC的近似Riemann解算子形式的数值流向量,将非线性方程转化为线性方程；在时间离散过程中,采用具有TVD性质的三阶Runge-Kutta时间离散方法。
(2)模型建立过程中,为了抑制间断处的非物理伪振荡,采用了间断检测器结合斜率限制器或稳定算子的方法。考虑了对底坡源项的离散处理,使模型能够处理复杂地形条件下包含强间断的浅水流动问题,并建立了一种均衡格式的Runge-Kutta间断有限元数值模型。
(3)将本文建立的Runge-Kutta间断有限元数值模型对浅水流动中的一些经典或存在解析解的算例,例如理想化溃坝问题、稳定水体微小扰动问题、跨临界流问题、水跃、潮波以及涌潮等问题,进行了数值模拟。而且在模拟过程中考虑了非棱柱体渠道、不连续底坡等实际情况。所有计算结果与解析解吻合较好,在间断处不含非物理的伪振荡,从而验证了模型的正确性与适用性。
The shallow water equations have a wide range of applications in the ocean, river, climate modeling, marine biology and environmental engineering.Related issues include storm surges, tsunamis and floods prediction, sediment and contaminant transport, etc.
A second-order Runge-Kutta discontinuous Galerkin finite element model is built based on one-dimensional shallow water equations, and is applied to numerical simulation of discontinuous shallow flows on irregular bottom topography successfully. The main contents and conclusions are as follows:
(1) The model is based on conservation forms of the shallow water equations. The computational domain is partitioned into a set of elements, and then the equations are integrated over the element. For the space discretization in the model, the HLL or HLLC approximate Riemann solver as the numerical flux at interfaces of elements is employed. For the time discretization, a third-order Runge-Kutta scheme which is TVD, is applied.
(2) In the model, a slope limiter or stabilization operator only around discontinuities using discontinuity detector is used in order to alleviate spurious oscillations near discontinuities. A discretization of the source terms is applied, so the model can solve the discontinuous shallow water flows with complex bed topography. So a well-balanced discontinuous Galerkin finite element model is constructed finally.
(3) The model is applied to numerical simulation of some classic discontinuous flow examples to prove its validity and applicability. The examples include idealized dam-break problem, a small perturbation of a steady-state water, transcritical flow over a hump, hydraulic jump, tidal waves and tidal bores. When solving discontinuous shallow water flow, the situations of non-prismatic channels and discontinuous bed topography can be considered. All the computations are good agreement with the analytical solutions, and the results exclude non-physical spurious oscillations near the shock.
本文在一维浅水方程组的基础上建立了二阶Runge-Kutta司断有限元数值模型,并成功应用于复杂地形条件下间断浅水流动的数值模拟。本文的主要内容和结论如下：
(1)从守恒形式的浅水方程组出发,对求解区域进行单元剖分,并在单元上对方程进行积分。在模型的空间离散过程中,跨单元边界处采用HLL和HLLC的近似Riemann解算子形式的数值流向量,将非线性方程转化为线性方程；在时间离散过程中,采用具有TVD性质的三阶Runge-Kutta时间离散方法。
(2)模型建立过程中,为了抑制间断处的非物理伪振荡,采用了间断检测器结合斜率限制器或稳定算子的方法。考虑了对底坡源项的离散处理,使模型能够处理复杂地形条件下包含强间断的浅水流动问题,并建立了一种均衡格式的Runge-Kutta间断有限元数值模型。
(3)将本文建立的Runge-Kutta间断有限元数值模型对浅水流动中的一些经典或存在解析解的算例,例如理想化溃坝问题、稳定水体微小扰动问题、跨临界流问题、水跃、潮波以及涌潮等问题,进行了数值模拟。而且在模拟过程中考虑了非棱柱体渠道、不连续底坡等实际情况。所有计算结果与解析解吻合较好,在间断处不含非物理的伪振荡,从而验证了模型的正确性与适用性。
The shallow water equations have a wide range of applications in the ocean, river, climate modeling, marine biology and environmental engineering.Related issues include storm surges, tsunamis and floods prediction, sediment and contaminant transport, etc.
A second-order Runge-Kutta discontinuous Galerkin finite element model is built based on one-dimensional shallow water equations, and is applied to numerical simulation of discontinuous shallow flows on irregular bottom topography successfully. The main contents and conclusions are as follows:
(1) The model is based on conservation forms of the shallow water equations. The computational domain is partitioned into a set of elements, and then the equations are integrated over the element. For the space discretization in the model, the HLL or HLLC approximate Riemann solver as the numerical flux at interfaces of elements is employed. For the time discretization, a third-order Runge-Kutta scheme which is TVD, is applied.
(2) In the model, a slope limiter or stabilization operator only around discontinuities using discontinuity detector is used in order to alleviate spurious oscillations near discontinuities. A discretization of the source terms is applied, so the model can solve the discontinuous shallow water flows with complex bed topography. So a well-balanced discontinuous Galerkin finite element model is constructed finally.
(3) The model is applied to numerical simulation of some classic discontinuous flow examples to prove its validity and applicability. The examples include idealized dam-break problem, a small perturbation of a steady-state water, transcritical flow over a hump, hydraulic jump, tidal waves and tidal bores. When solving discontinuous shallow water flow, the situations of non-prismatic channels and discontinuous bed topography can be considered. All the computations are good agreement with the analytical solutions, and the results exclude non-physical spurious oscillations near the shock.
引文
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[3]Zienkiewicz O C, Phillips D V. An automatic mesh generation scheme for plane and curved surface by isoparametric coordinates. Int. J. Num. Meth. Eng.,1971,3: 519-528
[4]Zieniewicz O C, Gago J P De S R, Kelly D W. The hierarchical concept in finite element analysis. Computers & structures,1983,16(1-4):53-65
[5]Zienkiewicz O C, Zhu J Z, Gong N G. Effective and practical h-p version adaptive analysis procedures for the finite element method. Int. J. Numer. Meth. Eng.,1989,28(1-6):879-891
[6]忻孝康,刘儒勋,蒋伯诚.计算流体动力学.长沙：国防科技大学出版社,1989
[7]Noelle S, Pankratz N, Puppo G, et al. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. Journal of Computational Physics,2006,213:474-499
[8]Reed W H, Hill T R. Triangular mesh methods for the neutron transport equation. LA Report,1973
[9]LeSaint P, Raviart P L. On a finite element method for solving the neutron transport equation. Mathematical aspect of finite element in partial differential equation, Academic Press,1974,89-145
[10]Chavent G, Salzano G. A finite element method for the 1d water flooding problem with gravity. J. Comput. Phys.,1982,45:307-344
[11]Chavent G, Cockburn B. The local projection P0-P1-discontinuous Galerkin method for scalar conservation laws. Model. Math. Anal. Numer.,1989,23: 565-592
[12]van Leer B. Towards the ultimate conservative difference scheme.Ⅱ. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys.,1974,14:361-370
[13]Cockburn B, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws Ⅱ:general framework. Mathematics of Computation.,1989,52(186):411-435
[14]Cockburn B, Lin S, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ:one dimensional systems. J. Comput. Phys.,1989,84(1):90-113
[15]Cockburn B, Hou S, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws Ⅳ:The multidimensional case. Math. Comp.,1990,54:545-581
[16]Cockburn B, Shu C W. The Runge-Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. ESAIM:Mathematical Modelling and Numerical Analysis.,1991,25(3):337-361
[17]Cockburn B, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V:Multidimensional systems. J. Comput. Phys.,1998,141:199-224
[18]Cockburn B, Shu C W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Scien. Comput.,2001,16(3):173-261
[19]Casulli V. Semi-implicit finite difference methods for the two-dimensional shallow water equation. J. Comput. Phys.,1990,86:56-74
[20]Grammeltvedt A. A survey of finite-difference schemes for the primitive equations for a barotropic fluid. Mon. Wea. Rev.,1969,97(5):384-404
[21]Castro M J, Garcia J A, Gonzalez J M, et al. Solving shallow-water systems in 2D domains using finite volume methods and multimedia SSE instructions. J. Comput. Appl. Math.,2008,221:16-32
[22]Gallouet T, Herard J M, Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput.& Fluids,2003,32: 479-513
[23]Toro E F, Garcia-Navarro P. Godunov-type methods for free-surface shallow flows:a review. J. Hydra. Res.,2007,45(6):737-751
[24]Hirsch C. Numerical computation of internal and external flows, computational methods for inviscid and viscous flows. Wiley:Chichester, England, New York, 1990
[25]Xing Y, Shu C W. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys.,2006,214:567-598
[26]Shu C W, Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys.,1988,77(2):439-471
[27]Cockburn B. Discontinuous Galerkin methods. Zeitschrift fur Angewandte Mathematik und Mechanik.,2003,83(11):731-754
[28]Schwanenberg D, Liem R, Kongeter J. Discontinuous Galerkin method for the shallow water equations.http://www.iww.rwth-aachen.de/fileadmin/internet/ research/papers/ShallowWaterEquations.pdf,2000
[29]Li H, Liu R. The discontinuous Galerkin finite element method for the 2D shallow water equations. Math. Comput. Simulation.,2001,56:223-233
[30]Aizinger V, Dawson C N. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv. WaterResour.,2002,25:67-84
[31]王立辉,胡四一.一维浅水方程组的间断有限元计算.水力发电学报,2007,26(1)：37～41
[32]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. Comput. Phys.,2001, 168:1-25
[33]Xing Y, Shu C W. A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commu. Comput. Phys.,2006,1(1):101-135
[34]Audusse E, Bouchut F, Bristeau M-O, et al. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput.,2004,25:2050-2065
[35]Kesserwani G, Ghostine R, Vazquez J, et al. Application of a second-order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms. Int. J. Numer. Meth. Fluids.,2008,56:805-821
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