应用间断有限元方法模拟一维非饱和土壤水流问题
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摘要
土壤水在水循环过程中起着重要作用,它和农业、水文学、环境科学密切相关。由于地下水流过程的复杂性,我们通常使用数值方法对地下水流问题进行模拟和预测。为了数值求解刻画一维饱和-非饱和土壤水流问题的非线性偏微分方程,标准Galerkin有限元方法是常见的选择。但对于解空间变化迅速的情形,标准有限元方法求解时需要较大程度的离散化,这往往超出实际可承受的范围。另一方面,椭圆或抛物方程的边值问题中一阶对流项主导二阶扩散项时,标准有限元方法进行网格细分导致自由度增加,却未必提高计算精度。Richards方程是广泛用于刻画渗流区域水流运动的模型。在某些特定条件下,Richards方程的解在时空上呈现陡峭的锋面。能否利用惩罚形式的间断有限元方法求解对流占优的非饱和土壤水流问题是本文研究的核心内容。本文首先将这种方法应用到Dirichlet边界条件下的Richards方程,采用内部惩罚间断有限元(IPDG)方法模拟特定条件下的土壤水分入渗。
在第一章,我们提出了本文研究的背景和意义,指出有限元方法在对流占优水流问题进行数值模拟时所面临的主要难点,对国内外间断有限元方法数值求解的研究进展进行了回顾和总结,阐述了间断有限元方法在模拟对流占优水流问题的优势,基于已有的研究成果和实践的需要,导出了本文的研究目标、研究内容和方法。
在第二章,我们介绍了Richards方程的两种形式,并给出标准有限元方法求解Richards方程的数值格式和内部惩罚间断有限元方法求解Richards方程的数值格式。与传统有限元方法不同,DG方法的基函数可以选为一组间断的正交基,本文标准有限元方法采用连续线性基函数,间断有限元方法采用分片线性基函数。间断有限元空间离散是从“上游”到“下游”逐层逐单元进行计算。我们给出了非稳定流问题的迭代公式,以经典的Warrick算例对间断有限元算法进行数值检验。
在第三章,IPDG方法和标准有限元方法同时被用于求解Dirichlet边界条件的Richards方程,本构关系采用van Genuchten-Mualem模型。我们根据美国USDA土壤质地分类选取12种土壤质地的均匀土壤剖面,不同剖分网格和初边值条件下的一维非稳定非饱和入渗算例对间断有限元算法进行检验。相对L2模和最大模误差用于检验算法的精度和有效性。数值模拟结果表明：在几种不同网格剖分单元上,IPDG方法可以有效的模拟均质砂土和壤砂土剖面的对流占优非饱和水流问题,相比FEM方法,DG方法的数值解具有更高的精度。除了能够很好地模拟解的剧烈改变,DG方法的数值结果呈现出很好的全局质量守恒性。之后我们给出DG方法表现一般和不理想的算例结果。
在论文的最后一章,我们进一步地总结了本文获得的研究成果,指出了目前研究的局限性和不足。
Soil water plays important role in water cycle, and closely related to the agriculture, hydrology, environment. Because of the complexity of the process of the groundwater flow process, it is usually necessary to simulate or predict the groundwater flow problem approximately using numerical techniques. Standard Galerkin method is always presented to solve the numerical solutions of the non-linear partial differential equations describing the simultaneous movement of water and solutes in a one-dimensional saturated-unsaturated problem. When applied to problems having rapidly changing solutions in the space variables, standard Galerkin method can often lead to unsatisfactory approximations at a practical or economically feasible level of discretization. In the case of first order or transport terms tend dominating the second order or diffusion terms, using of spaces with greater freedom does not necessarily produce better approximations of the exact solutions of elliptic and parabolic boundary problems. Richard's equation is the most common model used to describe water flow in the vadose zone, which can yield solutions with sharp fronts in space and time under certain conditions. The purpose of this thesis is to discuss whether and how to solve unsaturated convection-dominated diffusion problem in porous media by the interior penalty discontinuous Galerkin method. We use this method to solve a class of Richards equation for simulating specific condition soil water infiltration.
In chapter one, background of the research and significance of the paper are stated,and.We point out the main difficulty in convection-dominated diffusion problem of the water flow using finite element method.Related previous research work about discontinuous Galerkin method and interior penalty discontinuous Galerkin method for convection-diffusion problems in porous media. Based on previous literature and practical conditions, we set our research objectives, contents and methods to be used in this paper.
In chapter two, based on the idea of discontinuous Galerkin method, interior penalty discontinuous Galerkin (IPDG) method is applied to simulate one-dimension unsaturated infiltration problem. IPDG method and finite elements method (FEM) are both presented to solve the Richards equation with van Genuchten-Mualem model and Dirichlet conditions. The difference between DG and FEM is basic function. In this paper, linear polynomial is used as the basic function of FEM, and piecewise linear polynomial for DG method. The numerical solution can be developed from "upstream" to "downstream" along the streamline.The iteration formulas of transient flow problems have been deduced,then classic numerical sample of Warrick is used to test our IPDG method.
In chapter three, relative L2and maximum norms of the error are established. Twelve different kinds of soil profiles are presented and the numerical results show that, on several grids DG method could effectively simulate the unsaturated water flow in the specific soils for convection-dominated problem. DG solution can excellently approximate to the exact solution. The numerical experiments also demonstrated that for sand and loamy sand examples DG mehod could achieve accurate global mass balance. Otherwise,some acceptable results are discussed.
Lastly, in chapter four, we summarized some main conclusions for the chapters mentioned above, and presented the limitations of this research.
在第一章,我们提出了本文研究的背景和意义,指出有限元方法在对流占优水流问题进行数值模拟时所面临的主要难点,对国内外间断有限元方法数值求解的研究进展进行了回顾和总结,阐述了间断有限元方法在模拟对流占优水流问题的优势,基于已有的研究成果和实践的需要,导出了本文的研究目标、研究内容和方法。
在第二章,我们介绍了Richards方程的两种形式,并给出标准有限元方法求解Richards方程的数值格式和内部惩罚间断有限元方法求解Richards方程的数值格式。与传统有限元方法不同,DG方法的基函数可以选为一组间断的正交基,本文标准有限元方法采用连续线性基函数,间断有限元方法采用分片线性基函数。间断有限元空间离散是从“上游”到“下游”逐层逐单元进行计算。我们给出了非稳定流问题的迭代公式,以经典的Warrick算例对间断有限元算法进行数值检验。
在第三章,IPDG方法和标准有限元方法同时被用于求解Dirichlet边界条件的Richards方程,本构关系采用van Genuchten-Mualem模型。我们根据美国USDA土壤质地分类选取12种土壤质地的均匀土壤剖面,不同剖分网格和初边值条件下的一维非稳定非饱和入渗算例对间断有限元算法进行检验。相对L2模和最大模误差用于检验算法的精度和有效性。数值模拟结果表明：在几种不同网格剖分单元上,IPDG方法可以有效的模拟均质砂土和壤砂土剖面的对流占优非饱和水流问题,相比FEM方法,DG方法的数值解具有更高的精度。除了能够很好地模拟解的剧烈改变,DG方法的数值结果呈现出很好的全局质量守恒性。之后我们给出DG方法表现一般和不理想的算例结果。
在论文的最后一章,我们进一步地总结了本文获得的研究成果,指出了目前研究的局限性和不足。
Soil water plays important role in water cycle, and closely related to the agriculture, hydrology, environment. Because of the complexity of the process of the groundwater flow process, it is usually necessary to simulate or predict the groundwater flow problem approximately using numerical techniques. Standard Galerkin method is always presented to solve the numerical solutions of the non-linear partial differential equations describing the simultaneous movement of water and solutes in a one-dimensional saturated-unsaturated problem. When applied to problems having rapidly changing solutions in the space variables, standard Galerkin method can often lead to unsatisfactory approximations at a practical or economically feasible level of discretization. In the case of first order or transport terms tend dominating the second order or diffusion terms, using of spaces with greater freedom does not necessarily produce better approximations of the exact solutions of elliptic and parabolic boundary problems. Richard's equation is the most common model used to describe water flow in the vadose zone, which can yield solutions with sharp fronts in space and time under certain conditions. The purpose of this thesis is to discuss whether and how to solve unsaturated convection-dominated diffusion problem in porous media by the interior penalty discontinuous Galerkin method. We use this method to solve a class of Richards equation for simulating specific condition soil water infiltration.
In chapter one, background of the research and significance of the paper are stated,and.We point out the main difficulty in convection-dominated diffusion problem of the water flow using finite element method.Related previous research work about discontinuous Galerkin method and interior penalty discontinuous Galerkin method for convection-diffusion problems in porous media. Based on previous literature and practical conditions, we set our research objectives, contents and methods to be used in this paper.
In chapter two, based on the idea of discontinuous Galerkin method, interior penalty discontinuous Galerkin (IPDG) method is applied to simulate one-dimension unsaturated infiltration problem. IPDG method and finite elements method (FEM) are both presented to solve the Richards equation with van Genuchten-Mualem model and Dirichlet conditions. The difference between DG and FEM is basic function. In this paper, linear polynomial is used as the basic function of FEM, and piecewise linear polynomial for DG method. The numerical solution can be developed from "upstream" to "downstream" along the streamline.The iteration formulas of transient flow problems have been deduced,then classic numerical sample of Warrick is used to test our IPDG method.
In chapter three, relative L2and maximum norms of the error are established. Twelve different kinds of soil profiles are presented and the numerical results show that, on several grids DG method could effectively simulate the unsaturated water flow in the specific soils for convection-dominated problem. DG solution can excellently approximate to the exact solution. The numerical experiments also demonstrated that for sand and loamy sand examples DG mehod could achieve accurate global mass balance. Otherwise,some acceptable results are discussed.
Lastly, in chapter four, we summarized some main conclusions for the chapters mentioned above, and presented the limitations of this research.
引文
[1]Abdulle, A. Multiscale method based on discontinuous Galerkin methods for homogenization problems. Comptes Rendus Mathematique,2008,346 (1-2):97-102
[2]Arnold, D. N. An interior penalty finite element method with discontinuous elements. SIAM journal on numerical analysis,1982,19 (4):742-760
[3]Baba, K. and M. Tabata. On a conservation upwind finite element scheme for convective diffusion equations. ESAIM:Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique,1981,15 (1):3-25
[4]Bassi, F. and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics,1997, 131 (2):267-279
[5]Castillo, P. and B. Cockburn, et al. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM Journal on Numerical Analysis:2001,1676-1706
[6]Cockburn, B. and C. W. Shu TVB. Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅱ:general framework." Mathematics of Computation,1989, 411-435
[7]Cockburn, B. and C. W. Shu. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Mathematical Modelling and Numerical Analysis,1991,25:337-361
[8]Cockburn, B. and C. W. Shu. The local discontinuous galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis 1998,35 (6):2440-2463
[9]Cockburn, B. and C. W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems. Journal of Computational Physics,1998,141 (2):199-224
[10]Cockburn, B. and G. Kanschat, et al. Local discontinuous Galerkin methods for the Stokes system." SIAM Journal on Numerical Analysis:,2003,319-343
[11]Cockburn, B. and G. Kanschat, et al. The local discontinuous Galerkin method for linearized incompressible fluid flow:a review. Computers and Fluids,2005,34 (4-5):491-506
[12]Cockburn, B. and S. Hou, et al. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. Ⅳ:The multidimensional case. Mathematics of Computation,1990,545-581
[13]Cockburn, B. and S. Y. Lin, et al. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ:one-dimensional systems. J,1989, Comput. Phys 84 (1): 90-113
[14]Dosopoulos S, Lee J F. A parallel and non-conformal Interior Penalty Discontinuous Galerkin method for the time-domain Maxwell's equations. Antennas and Propagation (APSURSI),2011 IEEE. International Symposium on. IEEE,2011,3257-3260
[15]Douglas, J. J. and T. F. Russell. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM Journal on Numerical Analysis,1982,19 (5):871-885
[16]Efendiev, Y. and T. Hou. Multiscale finite element methods for porous media flows and their applications. Applied numerical mathematics,2007,57 (5-7):577-596
[17]Falk, R. S. and G R. Richter. Analysis of a continuous finite element method for hyperbolic equations. SIAM journal on numerical analysis,1987,24 (2):257-278
[18]Guo, W. and M. Stynes. Finite element analysis of exponentially fitted lumped schemes for time-dependent convection-diffusion problems. Numerische Mathematik,1993,66 (1):347-371
[19]Guo, W. and M. Stynes. Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations. Applied numerical mathematics,1994,15 (4):375-393
[20]Hou, T. Y. and X. H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics,1997,134 (1):169-189
[21]Houston, P. and I. Perugia, et al. Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM Journal on Numerical Analysis,2005,42 (1):434
[22]Hughes, T. J. and A. Brooks. A multidimensional upwind scheme with no crosswind diffusion." Finite element methods for convection dominated flows, AMD,1979,34:19-35
[23]Hughes, T. J. and G Engel, et al. A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency. Lecture Notes in Computational Science and Engineering,2000,11:135-146
[24]Johnson, C. Numerical solution of partial differential equations by the finite element method, Courier Dover Publications.2012
[25]Johnson, C. and J. Pitk Ranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Mathematics of Computation,1986,46 (173):1-26
[26]Johnson, C. and J. Saranen. Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations." Mathematics of Computation,1986,47 (175):1-18
[27]Li, H. and M. W. Farthing, et al.. Adaptive local discontinuous Galerkin approximation to Richards' equation. Advances in Water Resources,2007,30 (9):1883-1901
[28]Li, H. and M. W. Farthing, et al. Local discontinuous Galerkin approximations to Richards' equation. Advances in Water Resources,2007,30 (3):555-575
[29]McBride, D. and M. Cross, et al. Computational modelling of variably saturated flow in porous media with complex three-dimensional geometries. International journal for numerical methods in fluids,2006,50 (9):1085-1117
[30]Oden and T. J, et al. A discontinuous hp finite element method for diffusion problems,1998,146: 491-516
[31]Reed, W. H. and T. R. Hill. Triangular mesh methods for the neutron transport equation.1973
[32]Richards, L. A.. Capillary conduction of liquids through porous mediums. Journal of Applied Physics,2004,1 (5):318-333
[33]Richter, G R. An optimal-order error estimate for the discontinuous Galerkin method. Mathematics of Computation,1988,50 (181):75-88
[34]Richter, G. R.. The discontinuous Galerkin method with diffusion. Mathematics of computation, 1992,58 (198):631-643
[35]Roos, H. G Layer-Adapted Grids for Singular Perturbation Problems. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 1998,78 (5):291-309
[36]Song, L.. Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations. Numerical Methods for Partial Differential Equations,2012,28 (1):288-311
[37]Staff, S. S. D. Soil survey manual, United States Department of Agriculture,1993
[38]Sun, C. and S. J. Qin. The full discrete discontinuous finite element analysis for first-order linear hyperbolic equation. Journal of Computational Mathematics-International Edition,1999,17: 97-112
[39]Sun, S. and M. F. Wheeler. Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media." SIAM Journal on Numerical Analysis,2006,43 (1):195-219
[40]Van Genuchten, M. T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal,1980,44 (5):892-898
[41]Wang, H. F. and M. P. Anderson. Introduction to groundwater modeling. Finite difference and finite element methods:San Francisco,1982, WH Freeman 237
[42]Warrick, A. W. and D. O. Lomen, et al. A generalized solution to infiltration." Soil Science Society of America Journal,1985,49 (1):34
[43]Warrick, A. W. and J. W. Biggar, et al. Simultaneous solute and water transfer for an unsaturated soil.1971
[44]Wheeler, M. F. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis,1978,15 (1):152-161
[45]Xie, Z. and Z. Zhang. Superconvergence of DG Method for one-dimensional Singularly Perturbed Problems. Journal of Computational Mathematics,2007,25 (2)
[46]Xie, Z. and Z. Zhang, et al. A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems. Journal of Computational Mathematics,2009,27
[47]Yuan, L. and C. W. Shu. Discontinuous Galerkin method for a class of elliptic multi-scale problems. International Journal for Numerical Methods in Fluids,2008,56 (8):1017-1032
[48]Zhang, Z. and Z. Xie, et al. Superconvergence of discontinuous Galerkin methods for convection-diffusion problems. Journal of Scientific Computing,2009,41 (1):70-93
[49]黄明游On the Generalized Upwind Scheme for the Neutron Transport Equation.数学研究与评论,1985,4：013
[50]雷志栋与胡和平.土壤水研究进展与评述.水科学进展1999,10(003)：311-318
[51]雷志栋与杨诗秀等.土壤水动力学.北京：清华大学出版社.1988
[52]李荣华.偏微分方程数值解法,高等教育出版社.2005
[53]李荣华与陈仲英.微分方程广义差分法,吉林大学出版社,1994
[54]张铁.间断有限元理论与方法.北京：科学出版社.2012
[55]张铁与王宝艳.解对流占优反应扩散问题一致稳定的差分格式.高校应用数学学报A辑2010,02)：195-201
[2]Arnold, D. N. An interior penalty finite element method with discontinuous elements. SIAM journal on numerical analysis,1982,19 (4):742-760
[3]Baba, K. and M. Tabata. On a conservation upwind finite element scheme for convective diffusion equations. ESAIM:Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique,1981,15 (1):3-25
[4]Bassi, F. and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics,1997, 131 (2):267-279
[5]Castillo, P. and B. Cockburn, et al. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM Journal on Numerical Analysis:2001,1676-1706
[6]Cockburn, B. and C. W. Shu TVB. Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅱ:general framework." Mathematics of Computation,1989, 411-435
[7]Cockburn, B. and C. W. Shu. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Mathematical Modelling and Numerical Analysis,1991,25:337-361
[8]Cockburn, B. and C. W. Shu. The local discontinuous galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis 1998,35 (6):2440-2463
[9]Cockburn, B. and C. W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems. Journal of Computational Physics,1998,141 (2):199-224
[10]Cockburn, B. and G. Kanschat, et al. Local discontinuous Galerkin methods for the Stokes system." SIAM Journal on Numerical Analysis:,2003,319-343
[11]Cockburn, B. and G. Kanschat, et al. The local discontinuous Galerkin method for linearized incompressible fluid flow:a review. Computers and Fluids,2005,34 (4-5):491-506
[12]Cockburn, B. and S. Hou, et al. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. Ⅳ:The multidimensional case. Mathematics of Computation,1990,545-581
[13]Cockburn, B. and S. Y. Lin, et al. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ:one-dimensional systems. J,1989, Comput. Phys 84 (1): 90-113
[14]Dosopoulos S, Lee J F. A parallel and non-conformal Interior Penalty Discontinuous Galerkin method for the time-domain Maxwell's equations. Antennas and Propagation (APSURSI),2011 IEEE. International Symposium on. IEEE,2011,3257-3260
[15]Douglas, J. J. and T. F. Russell. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM Journal on Numerical Analysis,1982,19 (5):871-885
[16]Efendiev, Y. and T. Hou. Multiscale finite element methods for porous media flows and their applications. Applied numerical mathematics,2007,57 (5-7):577-596
[17]Falk, R. S. and G R. Richter. Analysis of a continuous finite element method for hyperbolic equations. SIAM journal on numerical analysis,1987,24 (2):257-278
[18]Guo, W. and M. Stynes. Finite element analysis of exponentially fitted lumped schemes for time-dependent convection-diffusion problems. Numerische Mathematik,1993,66 (1):347-371
[19]Guo, W. and M. Stynes. Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations. Applied numerical mathematics,1994,15 (4):375-393
[20]Hou, T. Y. and X. H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics,1997,134 (1):169-189
[21]Houston, P. and I. Perugia, et al. Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM Journal on Numerical Analysis,2005,42 (1):434
[22]Hughes, T. J. and A. Brooks. A multidimensional upwind scheme with no crosswind diffusion." Finite element methods for convection dominated flows, AMD,1979,34:19-35
[23]Hughes, T. J. and G Engel, et al. A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency. Lecture Notes in Computational Science and Engineering,2000,11:135-146
[24]Johnson, C. Numerical solution of partial differential equations by the finite element method, Courier Dover Publications.2012
[25]Johnson, C. and J. Pitk Ranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Mathematics of Computation,1986,46 (173):1-26
[26]Johnson, C. and J. Saranen. Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations." Mathematics of Computation,1986,47 (175):1-18
[27]Li, H. and M. W. Farthing, et al.. Adaptive local discontinuous Galerkin approximation to Richards' equation. Advances in Water Resources,2007,30 (9):1883-1901
[28]Li, H. and M. W. Farthing, et al. Local discontinuous Galerkin approximations to Richards' equation. Advances in Water Resources,2007,30 (3):555-575
[29]McBride, D. and M. Cross, et al. Computational modelling of variably saturated flow in porous media with complex three-dimensional geometries. International journal for numerical methods in fluids,2006,50 (9):1085-1117
[30]Oden and T. J, et al. A discontinuous hp finite element method for diffusion problems,1998,146: 491-516
[31]Reed, W. H. and T. R. Hill. Triangular mesh methods for the neutron transport equation.1973
[32]Richards, L. A.. Capillary conduction of liquids through porous mediums. Journal of Applied Physics,2004,1 (5):318-333
[33]Richter, G R. An optimal-order error estimate for the discontinuous Galerkin method. Mathematics of Computation,1988,50 (181):75-88
[34]Richter, G. R.. The discontinuous Galerkin method with diffusion. Mathematics of computation, 1992,58 (198):631-643
[35]Roos, H. G Layer-Adapted Grids for Singular Perturbation Problems. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 1998,78 (5):291-309
[36]Song, L.. Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations. Numerical Methods for Partial Differential Equations,2012,28 (1):288-311
[37]Staff, S. S. D. Soil survey manual, United States Department of Agriculture,1993
[38]Sun, C. and S. J. Qin. The full discrete discontinuous finite element analysis for first-order linear hyperbolic equation. Journal of Computational Mathematics-International Edition,1999,17: 97-112
[39]Sun, S. and M. F. Wheeler. Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media." SIAM Journal on Numerical Analysis,2006,43 (1):195-219
[40]Van Genuchten, M. T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal,1980,44 (5):892-898
[41]Wang, H. F. and M. P. Anderson. Introduction to groundwater modeling. Finite difference and finite element methods:San Francisco,1982, WH Freeman 237
[42]Warrick, A. W. and D. O. Lomen, et al. A generalized solution to infiltration." Soil Science Society of America Journal,1985,49 (1):34
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