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

• Published in: Transport in porous media Vol. 83; no. 3; pp. 637 - 652
• Main Authors: Juncu, Gheorghe, Nicola, Aurelian, Popa, Constantin
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2010; Springer; Springer Nature B.V
• Subjects: Civil Engineering; Classical and Continuum Physics; Computer simulation; Discretization; Earth and Environmental Science; Earth Sciences; Earth, ocean, space; Engineering and environment geology. Geothermics; Exact sciences and technology; Finite difference method; Flow equations; Geotechnical Engineering & Applied Earth Sciences; Hydrocarbons; Hydrogeology; Hydrology. Hydrogeology; Hydrology/Water Resources; Industrial Chemistry/Chemical Engineering; Infiltration; Iterative methods; Mathematical analysis; Mathematical models; Media; Multigrid methods; Nonlinearity; Numerical methods; Picard iterations; Pollution, environment geology; Porous media; Sedimentary rocks; Soil properties; Soil water; Two dimensional; Unsaturated; Unsaturated flow; Unsaturated soils; Water flow; Finite difference; Unsaturated porous media; Richards’ equation; Nonlinear multigrid algorithm; algorithms; models; finite difference analysis; transport; water; Richards equation; infiltration; hydraulics; solution; soils; porous media; methodology; Richards' equation
• ISSN: 0169-3913; 1573-1634
• DOI: 10.1007/s11242-009-9465-3

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.