A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam
Abstract
:1. Introduction
2. Materials and Methods
2.1. Richards’ Equation
2.2. Finite Difference Formulation
- If the grid is regular, for instance, an actual rectangular grid, the matrix coefficient in Equation (9) will be equal for all the inner grid nodes. Additionally, if the right-hand side of that equation is constant, which is the case in several important equations (v.gr. Poisson equation), the coefficients will be the same for all the inner grid nodes. This implies that in specific modeling applications, the calculation of the coefficients can be very efficient.
- Furthermore, if the positions of the neighbors are symmetrical to the central node and the differential operator is self-adjoint, the number of different coefficients required decreases (v.gr. four nodes is the discretization of the Poisson equation). This is precisely the case of the standard differences in rectangular grids. However, the analogous case occurs in other regular structures, such as those used in [39].
2.3. Case of Study
2.3.1. Problem Domain and Boundary Conditions
2.3.2. Grid Independence
2.3.3. Stationary Flow in a Dam
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Short Biography of Author
Carlos Chávez-Negrete is a professor of in the Civil Engineering School of the Universidad Michoacana de San Nicolás de Hidalgo (UMSNH) in Morelia, Mexico. He holds a Ph.D. in Geotechnical Engineering from Universitat Politècnica de Catalunya, in Spain. His research interest includes laboratory characterization of soils and rocks, slope stability analysis, pavement analysis, and design, the development of numerical methods for geotechnical engineering, and unsaturated soil mechanics. | |
Francisco Domínguez-Mota completed a B.Sc. in Physics and Mathematics at the Universidad Michoacana in México. He obtained a Master degree in Applied Mathematics from the Center of Research in Mathematics in Guanajuato, Mexico, and a Ph.D in Mathematics at the Universidad Nacional Autónoma de México. He is a member of the National System of Researchers (SNI) in México, and is a researcher in Applied Mathematics at the Universidad Michoacana. His areas of research include numerical solution of differential equations and applications in engineering, numerical generation of structured grids and applications of optimization in large scale problems. | |
Daniel Santana-Quinteros is a M.Sc. student at the Centro de ciencias matemáticas UNAM campus Morelia. He completed a B.Sc. in Physics and Mathematic at the Universidad Michoacana in México. He has been a math teacher since 2018. His areas of research include numerical solution of differential equations, flow in porous media, biomathematics and systems biology. |
Mesh | A | B |
---|---|---|
Mesh 0 | 0.03572126729978147 | 0.09399951295707337 |
Mesh 1 | 0.00629495397038204 | 0.01468723574184271 |
Mesh 2 | 0.00213435762711308 | 0.00213435762711308 |
Parameter | Value |
---|---|
0.05 | |
0.5 | |
(1/kPa) | 0.86 |
n | 1.57 |
ks (m/s) |
Method | Mesh | LS Slope | Order | |
---|---|---|---|---|
GFDM | 1 | 0.25 | −0.0211 | 0.9843 |
GFDM | 1 | 0.50 | −0.0341 | 0.9947 |
GFDM | 1 | 0.75 | −0.0486 | 0.9974 |
GFDM | 1 | 1.00 | −0.0642 | 0.9989 |
FEM | 1 | 0.25 | −0.1017 | 1.0203 |
FEM | 1 | 0.50 | −0.2147 | 1.0041 |
FEM | 1 | 0.75 | −0.2172 | −1.0301 |
FEM | 1 | 1.00 | −0.1505 | 0.1884 |
GFDM | 2 | 0.25 | −0.1374 | 0.9978 |
GFDM | 2 | 0.50 | −0.2772 | 0.9952 |
GFDM | 2 | 0.75 | −0.4542 | 2.6186 |
GFDM | 2 | 1.00 | 0.0000 | −1.3084 |
FEM | 2 | 0.25 | −0.1180 | 0.9987 |
FEM | 2 | 0.50 | −0.2076 | 2.2392 |
FEM | 2 | 0.75 | −0.1362 | −2.6338 |
FEM | 2 | 1.00 | −0.0001 | −3.2192 |
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Chávez-Negrete, C.; Santana-Quinteros, D.; Domínguez-Mota, F. A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam. Mathematics 2021, 9, 1604. https://doi.org/10.3390/math9141604
Chávez-Negrete C, Santana-Quinteros D, Domínguez-Mota F. A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam. Mathematics. 2021; 9(14):1604. https://doi.org/10.3390/math9141604
Chicago/Turabian StyleChávez-Negrete, Carlos, Daniel Santana-Quinteros, and Francisco Domínguez-Mota. 2021. "A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam" Mathematics 9, no. 14: 1604. https://doi.org/10.3390/math9141604