Articles | Volume 21, issue 6
https://doi.org/10.5194/hess-21-2667-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/hess-21-2667-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation
Fadji Hassane Maina
Laboratoire d'Hydrologie et de Géochimie de Strasbourg, Univ. Strasbourg/EOST – CNRS, 1 rue Blessig, 67084 Strasbourg, France
CEA-Laboratoire de Modélisation des Transferts dans
l'Environnement, Bât. 225, 13108 Saint Paul lez Durance cedex, France
Philippe Ackerer
CORRESPONDING AUTHOR
Laboratoire d'Hydrologie et de Géochimie de Strasbourg, Univ. Strasbourg/EOST – CNRS, 1 rue Blessig, 67084 Strasbourg, France
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Cited
18 citations as recorded by crossref.
- A novel semi-numerical infiltration model combining conceptual and physically based approaches F. Stanić et al. https://doi.org/10.1016/j.jhydrol.2025.132664
- An elastoplastic model with egg-shaped yield surface for coastal soft clay L. Ju et al. https://doi.org/10.1016/j.apor.2024.103975
- A mass-conservative predictor-corrector solution to the 1D Richards equation with adaptive time control Z. Li et al. https://doi.org/10.1016/j.jhydrol.2020.125809
- Numerical modeling of one-dimensional variably saturated flow in a homogeneous and layered soil–water system via mixed form Richards equation with Picard iterative scheme S. Shah et al. https://doi.org/10.1007/s40808-022-01588-z
- An advanced discrete fracture model for variably saturated flow in fractured porous media B. Koohbor et al. https://doi.org/10.1016/j.advwatres.2020.103602
- Efficient mass conservative numerical model for solving variably saturated groundwater flow A. Aharmouch & B. Amaziane https://doi.org/10.1016/j.jhydrol.2021.126976
- Multistep optimization of HyPix model for flexible vertical scaling of soil hydraulic parameters J. Pollacco et al. https://doi.org/10.1016/j.envsoft.2022.105472
- Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils Y. Zha et al. https://doi.org/10.1002/wat2.1364
- A thermodynamics-based coupled model for multi-phase flowback in deformable dual-porosity shale gas reservoirs K. Wang et al. https://doi.org/10.1016/j.jhydrol.2024.132187
- A comparison of numerical schemes for the GPU-accelerated simulation of variably-saturated groundwater flow Z. Li et al. https://doi.org/10.1016/j.envsoft.2023.105900
- A Fully Coupled Hydro-Mechanical-Gas Model Based on Mixture Coupling Theory S. Abdullah et al. https://doi.org/10.1007/s11242-022-01784-6
- A finite element method using a bounded auxiliary variable for solving the Richards equation A. Benfanich et al. https://doi.org/10.1016/j.jcp.2026.115127
- Sensitivity of meteorological-forcing resolution on hydrologic variables F. Maina et al. https://doi.org/10.5194/hess-24-3451-2020
- Switching the Richards’ equation for modeling soil water movement under unfavorable conditions J. Zeng et al. https://doi.org/10.1016/j.jhydrol.2018.06.069
- A Reduced-Order Algorithm for a Digital Twin Model of Ultra-High-Voltage Valve-Side Bushing Considering Spatio-Temporal Non-Uniformity Y. He et al. https://doi.org/10.3390/en18061481
- Mathematical Analysis of a Subsurface Flow Model S. Al Nazer et al. https://doi.org/10.1137/22M1542076
- HyPix: 1D physically based hydrological model with novel adaptive time-stepping management and smoothing dynamic criterion for controlling Newton–Raphson step J. Pollacco et al. https://doi.org/10.1016/j.envsoft.2022.105386
- Numerical Solvers for the Richards Equation: A Comparative Review of Stability, Efficiency and Mass Conservation A. Chatzikamaris et al. https://doi.org/10.3390/eng7090467
18 citations as recorded by crossref.
- A novel semi-numerical infiltration model combining conceptual and physically based approaches F. Stanić et al. https://doi.org/10.1016/j.jhydrol.2025.132664
- An elastoplastic model with egg-shaped yield surface for coastal soft clay L. Ju et al. https://doi.org/10.1016/j.apor.2024.103975
- A mass-conservative predictor-corrector solution to the 1D Richards equation with adaptive time control Z. Li et al. https://doi.org/10.1016/j.jhydrol.2020.125809
- Numerical modeling of one-dimensional variably saturated flow in a homogeneous and layered soil–water system via mixed form Richards equation with Picard iterative scheme S. Shah et al. https://doi.org/10.1007/s40808-022-01588-z
- An advanced discrete fracture model for variably saturated flow in fractured porous media B. Koohbor et al. https://doi.org/10.1016/j.advwatres.2020.103602
- Efficient mass conservative numerical model for solving variably saturated groundwater flow A. Aharmouch & B. Amaziane https://doi.org/10.1016/j.jhydrol.2021.126976
- Multistep optimization of HyPix model for flexible vertical scaling of soil hydraulic parameters J. Pollacco et al. https://doi.org/10.1016/j.envsoft.2022.105472
- Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils Y. Zha et al. https://doi.org/10.1002/wat2.1364
- A thermodynamics-based coupled model for multi-phase flowback in deformable dual-porosity shale gas reservoirs K. Wang et al. https://doi.org/10.1016/j.jhydrol.2024.132187
- A comparison of numerical schemes for the GPU-accelerated simulation of variably-saturated groundwater flow Z. Li et al. https://doi.org/10.1016/j.envsoft.2023.105900
- A Fully Coupled Hydro-Mechanical-Gas Model Based on Mixture Coupling Theory S. Abdullah et al. https://doi.org/10.1007/s11242-022-01784-6
- A finite element method using a bounded auxiliary variable for solving the Richards equation A. Benfanich et al. https://doi.org/10.1016/j.jcp.2026.115127
- Sensitivity of meteorological-forcing resolution on hydrologic variables F. Maina et al. https://doi.org/10.5194/hess-24-3451-2020
- Switching the Richards’ equation for modeling soil water movement under unfavorable conditions J. Zeng et al. https://doi.org/10.1016/j.jhydrol.2018.06.069
- A Reduced-Order Algorithm for a Digital Twin Model of Ultra-High-Voltage Valve-Side Bushing Considering Spatio-Temporal Non-Uniformity Y. He et al. https://doi.org/10.3390/en18061481
- Mathematical Analysis of a Subsurface Flow Model S. Al Nazer et al. https://doi.org/10.1137/22M1542076
- HyPix: 1D physically based hydrological model with novel adaptive time-stepping management and smoothing dynamic criterion for controlling Newton–Raphson step J. Pollacco et al. https://doi.org/10.1016/j.envsoft.2022.105386
- Numerical Solvers for the Richards Equation: A Comparative Review of Stability, Efficiency and Mass Conservation A. Chatzikamaris et al. https://doi.org/10.3390/eng7090467
Saved (final revised paper)
Latest update: 27 Sep 2026
Short summary
In many fields like climate change, hydrology and agronomy, water movement in unsaturated soils is usually simulated using the Richards equation. However, this equation requires lot of computational effort to be solved due to its highly nonlinear behavior, which hampers its use in simulations. In this paper, we analyze and developed some numerical strategies and we evaluate their reliability and efficiency.
In many fields like climate change, hydrology and agronomy, water movement in unsaturated soils...