We discuss several time integration techniques of system of ordinary differential equations characterized by strong nonlinear coupling of the unknown variables. This system is a result of spatial discretization (using such as finite element, finite difference, or finite volume element) of the Richards Equation which is a governing mathematical principle for modeling water infiltration through a subsurface. The nature of Richards Equation is further complicated by the fact that the rate of

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... of the quantity of interest represented by a time derivative is also nonlinear. We formulate a general framework of the numerical time integration as a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in time variable to the variational equation and appropriate "variational crime" in the form of numerical quadrature for calculating the integration in the formulation. The resulting nonlinear algebraic equations are solved by employing some fixed point type iterations. We discuss two examples and compare their performance. Richards Equation Flow through porous media has been of interest to people in many fields of engineering, agricultural and chemical sciences. In many problems, a porous medium is occupied by more than one fluids that are immiscible, i.e., they are only slightly soluble or completely non-soluble. Consequently, each of these fluids have to be treated individually as different phases. Also the pressure of each phase is related by the capillary pressure. This is the basic reason for describing the flow in unsaturated zone as a multiphase system. The loose and solid matrices represent the spatially fixed subsystem, while the void volume ( sometimes referred to as pore space ) that contain gas and/or liquids represent the mobile subsystem whose movement/flow we want to analyze. It has been known that this system is subject to both spatial and temporal variations, and the complexity of the nature results in coupling of the physical and chemical processes. We are interested in modeling the flow of water into a porous medium whose pore space is filled with air and some water. Several terminologies are in order. The fraction of the pore space volume to the porous medium total volume is called porosity, which is denoted by φ. The amount of water filling in the pore space of the medium is represented by the water saturation, S , i.e., it is defined as the fraction of the total pore space that is filled with water. In this connection, we say that the saturation varies between two values, namely, the residual water saturation, S r , and the fully saturated value, S s . These parameters are specific to different porous medium. Another measure of the amount