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Keywords = nonlinear unsaturated soil water flow problem

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26 pages, 952 KiB  
Article
A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem
by Xiaoli Hou, Fei Teng, Zhendong Luo and Hui Fu
Mathematics 2024, 12(11), 1726; https://doi.org/10.3390/math12111726 - 1 Jun 2024
Cited by 1 | Viewed by 812
Abstract
The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design [...] Read more.
The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems. Full article
(This article belongs to the Section E: Applied Mathematics)
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16 pages, 346 KiB  
Article
Modeling Water Flow in Variably Saturated Porous Soils and Alluvial Sediments
by Mauro Giudici
Sustainability 2023, 15(22), 15723; https://doi.org/10.3390/su152215723 - 8 Nov 2023
Cited by 4 | Viewed by 1643
Abstract
The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial [...] Read more.
The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial differential equation, which translates the mass conservation principle into mathematical terms. Such an equation assumes that the variation of the volumetric water content (θ) in the medium is balanced by the net flux of water flow, i.e., the divergence of specific discharge, if source/sink terms are negligible. Specific discharge is in turn related to the matric potential (h), through the non-linear Darcy–Buckingham law. The resulting equation can be rewritten in different ways, in order to express it as a partial differential equation where a single physical quantity is considered to be a dependent variable. Namely, the most common instances are the Fokker–Planck Equation (for θ), and the Richards Equation (for h). The other two forms can be given for generalized matric flux potential (Φ) and for hydraulic conductivity (K). The latter two cases are shown to limit the non-linearity to multiplicative terms for an exponential K-to-h relationship. Different types of boundary conditions are examined for the four different formalisms. Moreover, remarks given on the physico-mathematical properties of the relationships between K, h, and θ could be useful for further theoretical and practical studies. Full article
(This article belongs to the Special Issue Groundwater, Soil and Sustainability)
21 pages, 4035 KiB  
Article
Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition
by Dariusz Gąsiorowski and Tomasz Kolerski
Water 2020, 12(6), 1780; https://doi.org/10.3390/w12061780 - 23 Jun 2020
Cited by 8 | Viewed by 4409
Abstract
Research on seepage flow in the vadose zone has largely been driven by engineering and environmental problems affecting many fields of geotechnics, hydrology, and agricultural science. Mathematical modeling of the subsurface flow under unsaturated conditions is an essential part of water resource management [...] Read more.
Research on seepage flow in the vadose zone has largely been driven by engineering and environmental problems affecting many fields of geotechnics, hydrology, and agricultural science. Mathematical modeling of the subsurface flow under unsaturated conditions is an essential part of water resource management and planning. In order to determine such subsurface flow, the two-dimensional (2D) Richards equation can be used. However, the computation process is often hampered by a high spatial resolution and long simulation period as well as the non-linearity of the equation. A new highly efficient and accurate method for solving the 2D Richards equation has been proposed in the paper. The developed algorithm is based on dimensional splitting, the result of which means that 1D equations can be solved more efficiently than as is the case with unsplit 2D algorithms. Moreover, such a splitting approach allows any algorithm to be used for space as well as time approximation, which in turn increases the accuracy of the numerical solution. The robustness and advantages of the proposed algorithms have been proven by two numerical tests representing typical engineering problems and performed for typical properties of soil. Full article
(This article belongs to the Section Hydrology)
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