A Stable Radial Basis Function Partition of Unity Method with d-Rectangular Patches for Modelling Water Flow in Porous Media

Abstract

Richards’ equation is used to describe water flow in porous media. It is strongly nonlinear, which makes the approximation of its solution a challenging task. In this study, we aim to linearize Richards’ equation using the exponential Gardner models of hydraulic conductivity and moisture content variables. Then, we propose a new formulation of radial basis function partition of unity method (RBFPUM) based on d-rectangular patches as local subdomains in order to approximate the solution of the linearized Richards’ equation. In particular, the multivariate weights are expressed as a product of one dimensional Wendland compactly supported radial basis functions. With small values of the shape parameter contained in the Gaussian radial basis function, safe computations of the solution will be achieved by adopting RBF-QR algorithm (Forenberg et al. in SIAM J Sci Comput 33(2):869–892, 2011). Hence, an efficient numerical solution can be obtained not only in terms of safe computations but also in terms of accuracy, as we will see in the numerical examples. In order to deal with layered soils, another method is proposed for the 2D and the 3D cases. It is based on the domain decomposition principle coupled with RBFPUM by using d-rectangular patches and the RBF-QR algorithm. The matching conditions, i.e. the continuity of mass fluxes and pressure head, are imposed at the interface between zones via the Steklov–Poincare equation. The efficiency of the proposed method will be examined via some examples.