Numerical solution of Richards' equation of water flow by generalized finite differences

Richards equation is a degenerate elliptic parabolic nonlinear expression which models flow in unsaturated porous media. Due to its importance in engineering, a number of linearization schemes for approximating its solution have been proposed. Among the more efficient are combinations of Newtonian iterations for the spatial discretization using finite elements, and an implicit θ-method for the time integration. However, when the finite element formulation is used, numerical oscillations near the infiltration front are presented. To overcome this problem, this paper presents a novel generalized finite differences scheme and an adaptive step size Crank-Nicolson method, which can be applied for solving Richards' equation on nonrectangular structured grids. The proposed method is tested on an illustrative numerical example on a road embankment and the results are compared with a finite element method solution.