Application Of The Discontinuous Galerkin Method To Simulate One-dimension Unsaturated Soil Water Flow Problems

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.