Application of RBF-DQ Method to Time-Dependent Analysis of Unsaturated Seepage

Richards’ equation is a nonlinear partial differential equation governing unsteady seepage flow through unsaturated porous media. This paper investigates applicability of radial basis function-based differential quadrature (RBF-DQ), as a meshless method, to simulate one-dimensional flow processes in the unsaturated zone under different initial and boundary conditions. Fourth-order Runge–Kutta scheme has been adopted for time integration. Results of solving three numerical examples using RBF-DQ are compared with those of analytical, numerical, and experimental solutions presented in the literature. The comparison indicates that RBF-DQ can provide more accurate results comparing with traditional FDM or FEM without the need to discretize the computational domain. Moreover, the merit of mesh-free characteristic in RBF-DQ makes it suitable not only for solving nonlinear problems but also for dealing with multidimensional problems since meshless methods are not restricted to dimensional limitations. A key parameter in utilizing multiquadratic approximation in RBF-DQ method is the user-defined shape parameter C, which may significantly affect solution accuracy. Thus, a sensitivity analysis has been conducted to study possible effects of shape parameter on achieved results.