Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation

Baron, Vincent; Coudière, Yves; Sochala, Pierre

doi:10.1016/j.apnum.2016.10.005

Cited by 13 publications

(12 citation statements)

References 28 publications

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“…We hope instead that there will be continued development and adoption of new spatial and temporal discretization schemes. Following trends of maturing computational fields, we expect to see improved, adaptive algorithms driven by solution dynamics and a posteriori error estimates (Mostaghimi et al, 2015;Baron et al, 2017). Moreover, approaches are needed that consider entire solution error and dynamics-for example, joint space-time discretization and error control (Solin and Kuraz, 2011) rather than just a MOL approach with temporal adaptivity but static spatial approximation (Farthing et al, 2003b).…”

Section: Discussion and Alternativesmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

257

138

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The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

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Section: Discussion and Alternativesmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

257

138

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“…In this subsection, we numerically investigate the rates of convergence in space and time of the different schemes. Following the framework presented in [7,70], we employ a manufactured solution to test the methods. The idea is to use an arbitrary sufficiently differentiable function as exact solution with a source term for the computation of numerical errors.…”

Section: Manufactured Solutionsmentioning

confidence: 99%

“…Here, we perform numerical simulations using the test case presented in [7] for two layers of soil with same characteristics given by the relations (5.13)-(5.14) for the capillary pressure and the relative hydraulic conductivity with the parameter values (5.19), except for the saturated hydraulic conductivity K s which changes in the medium. The computational domain is the square [0 cm, 100 cm]×[0 cm, 100 cm] with variable saturated hydraulic conductivity as shown in Figure 16.…”

Section: Layers Of Soil Of L-shape Formmentioning

confidence: 99%

Implicit and semi-implicit second-order time stepping methods for the Richards equation

Keita

Beljadid

Bourgault

2021

Advances in Water Resources

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This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study and a series of numerical tests are performed to analyze efficiency and robustness of the different schemes. The developed scheme, based on the proposed temporal extrapolation techniques and the mixed formulation involving the saturation and pressure head and using the standard linear Lagrange element, performs better than other schemes based on the saturation and the flux and using the Raviart-Thomas elements. The proposed semi-implicit scheme is a good alternative when implicit schemes meet convergence issues.

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“…Srivastava and Yeh (1991) used a Laplace transformation technique to obtain analytical solutions of the transient flows in homogeneous and two-layer soils, while Tracy (2006) proposed two-and three-dimensional analytical solutions of the RE using simple exponential models. Although their actual applicability is limited, these are good references for theoretical analysis (Zambra et al, 2012;Liu et al, 2015;Baron et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

Zhu

2021

Preprint

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The numerical solution of various systems of linear equations describing fluid infiltration uses the Picard iteration (PI). However, because many such systems are ill-conditioned, the solution process often has a poor convergence rate, making it very time-consuming. In this study, a control volume method based on non-uniform nodes is used to discretize the Richards equation, and adaptive relaxation is combined with a multistep preconditioner to improve the convergence rate of PI. The resulting adaptive relaxed PI with multistep preconditioner (MP(m)-ARPI) is used to simulate unsaturated flow in porous media. Three examples are used to verify the proposed schemes. The results show that MP(m)-ARPI can effectively reduce the condition number of the coefficient matrix for the system of linear equations. Compared with conventional PI, MP(m)-ARPI achieves faster convergence, higher computational efficiency, and enhanced robustness. These results demonstrate that improved scheme is an excellent prospect for simulating unsaturated flow in porous media.

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Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation

Cited by 13 publications

References 28 publications

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Implicit and semi-implicit second-order time stepping methods for the Richards equation

Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

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