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–Poincaré equation. The efficiency of the proposed method will be examined via some examples.

理查兹方程用于描述多孔介质中的水流。它是强非线性的，这使其解的近似成为一项艰巨的任务。在这项研究中，我们旨在使用水力传导率和水分含量变量的指数Gardner模型线性化Richards方程。然后，我们提出了基于d的统一方法径向基函数分区（RBFPUM）的新公式。-矩形补丁作为局部子域，以便近似线性化Richards方程的解。特别地，多元权重表示为一维Wendland紧密支持的径向基函数的乘积。使用高斯径向基函数中包含的形状参数较小的值，将通过采用RBF-QR算法实现安全的解计算（Forenberg等人，SIAM J Sci Comput 33（2）：869–892，2011） 。因此，正如我们将在数值示例中看到的那样，不仅可以在安全计算方面而且可以在精度方面获得有效的数值解。为了处理分层土壤，针对2D和3D情况提出了另一种方法。它基于域分解原理，结合RBFPUM，使用d-矩形补丁和RBF-QR算法。匹配条件，即通量和压头的连续性，通过Steklov-Poincaré方程施加在区域之间的界面上。将通过一些示例来检验所提出方法的效率。