Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

Abstract

Picard and Newton iterations are widely used to solve numerically the nonlinear Richards' equation (RE) governing water flow in unsaturated porous media. When solving RE in two space dimensions, direct methods applied to the linearized problem in the Newton/Picard iterations are inefficient. The numerical solving of RE in 2D with a nonlinear multigrid (MG) method that avoids Picard/Newton iterations is the focus of this work. The numerical approach is based on an implicit, second-order accurate time discretization combined with a second-order accurate finite difference spatial discretization. The test problems simulate infiltration of water in 2D unsaturated soils with hydraulic properties described by Broadbridge-White and van Genuchten-Mualem models. The numerical results show that nonlinear MG deserves to be taken into consideration for numerical solving of RE.

Differential Quadrature Method for Two-Dimensional Unsaturated Water Flow

“…For the 2D case, two examples in ve test cases are considered to investigate the e ectiveness of the scheme. The results are in close agreement with those obtained by multigrid (MG) [20] approach previously published in the literature.…”

“…The parameters' values as well as initial and boundary conditions used to simulate the three cases of this problem are as follows: The gures are in very close agreement with those obtained by using MG [20]. It is worthy to point out that MG converges to accurate results by employing a 64 256 grid.…”

“…In this study, it is convenient to write the moisture-content form of this equation for homogeneous and isotropic soils as the normalized form, as in the following [20]: [26] proposed the following relationships for hydraulic conductivity and di usivity: …”

“…Various types of boundary conditions, including Dirichlet, Neumann, and Robin boundary conditions, are also investigated. These test cases have been chosen since they have been already solved numerically using Multi Grid (MG) method by Juncu et al (2010) [20].…”

“…[15][16][17][18][19]). In addition to the aforementioned methods, other numerical approaches, such as multigrid method [20] and Backward Di erentiation Formula (BDF), combined with exponential time integration approach, [21] have been developed for the simulation of unsaturated ow.…”