Localized MQ-RBF meshless techniques for modeling unsaturated flow

Abstract

In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.