Space-time polyharmonic radial polynomial basis functions for modeling saturated and unsaturated flows

In this paper, we propose a novel meshless approach that involves using space-time polyharmonic radial polynomial basis functions for modeling saturated and unsaturated flows in porous media. In this study, space-time polyharmonic radial polynomial basis functions were developed in the space-time domain using a meshless collocation method. This domain contains three sets of collocation points, namely the inner, source, and boundary points, for the spatial and temporal discretization of the governing equation. Because the initial and boundary data are accessible space-time boundaries, the solutions of groundwater flows problems are approximated by solving the inverse boundary value problem in the space-time domain without using the conventional time-marching scheme. Saturated and unsaturated flow problems were investigated to demonstrate the robustness of the proposed method. The results obtained using the proposed approach were compared with those obtained using the conventional polyharmonic spline radial basis function. The proposed space-time polyharmonic radial polynomial basis functions obtained highly accurate solutions. Moreover, in solving saturated and unsaturated flow problems, the accuracy and stability of the proposed functions were higher than those of the conventional time-marching scheme.