A Mixed Element-Free Galerkin Method for the Richards' Equation

Abstract

In this paper, we introduce a mixed element-free Galerkin (MEFG) method to solve three-dimensional (3D) nonlinear Richards' equation. The 3D Richards' equation is first linearized by applying the Kirchhoff transform and transformed into a series of two-dimensional (2D) equations along the selected splitting direction, and then the improved element-free Galerkin (IEFG) method is used to solve the 2D initial boundary value problem with the finite difference method (FDM) in the splitting direction and the time domain. Finally, the numerical solution of the required 3D problem is obtained. We have selected some arithmetic cases about Richards' equation, and solved them with IEFG and MEFG method respectively. Through the comparative analyses of the calculation results of the two methods, we are able to see the validity and superiority of the MEFG in solving the 3D nonlinear Richards' equation.

Keywords: Richards' equation, nonlinear, mixed element-free Galerkin method, improved element-free Galerkin method