Locally conservative, stabilized finite element methods for variably saturated flow

Kees, Christopher E.; Farthing, Matthew W.; Dawson, Clint

doi:10.1016/j.cma.2008.06.005

Cited by 50 publications

(28 citation statements)

References 77 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[6] and [7] are only intended to provide some context for the following discussion, and we have neglected any details of the solution approximation or evaluation of the integrals in Eq. [6] and [7] (Kees et al, 2008).…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…As a result, cell-centered finite differences and low-order finite volumes/ integrated finite differences, and piecewise linear, continuous Galerkin finite element schemes remain the dominant schemes used in practice (Selker and John, 2004;Kollet and Maxwell, 2006;Yeh et al, 2011;Diersch, 2013). We note that post-processing is increasingly being used in conjunction with continuous Galerkin methods to provide velocity fields that are locally conservative over elemental control volumes (Larson and Niklasson, 2004;Sun and Wheeler, 2006;Kees et al, 2008;Scudeler et al, 2016).…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Since then, many authors have presented critiques of Richards' equation and existing solution techniques. A far-from-exhaustive sample includes (Forsyth et al, 1995;Tocci et al, 1997;van Dam and Feddes, 2000;Vogel et al, 2001;Kavetski et al, 2001a;Farthing et al, 2003b;Bause and Knabner, 2004;Manzini and Ferraris, 2004;D'Haese et al, 2007;Kees et al, 2008;Vogel and Ippisch, 2008;Juncu et al, 2011;Berninger et al, 2014;Lipnikov et al, 2016). Our intention here is not to repeat what those authors have written, rather it is to assess the state of the art, limitations, needs, and alternatives.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Farthing

Ogden

2017

Soil Science Soc of Amer J

249

138

View full text Add to dashboard Cite

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...

show abstract

Section: Spatial Discretizationmentioning

confidence: 99%

Section: Spatial Discretizationmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

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

Farthing

Ogden

2017

Soil Science Soc of Amer J

249

138

View full text Add to dashboard Cite

show abstract

“…For the case of saturated-unsaturated flow, we mention the contributions of Forsyth et al (1995), Pan and Wierenga (1995), Diersch and Perrochet (1999), Berganaschi and Putti (1999), Wiliams et al (2000), Lima-Vivancos and Voller (2004), Bause and Knabner (2004), Hao et al (2005), Marinoschi (2005), Basombrio et al (2006), McBride et al (2006), Pei et al (2006), Kees et al (2008), Bevilacqua et al (2009), Casulli and Zanolli (2010), Radu et al (2010), and Wu (2010). For 2D and 3D problems, the standard numerical approach consists of: -space discretization by finite volume, finite element (Galerkin), mixed finite element, finite differences; -method of lines for time discretization; -solution of the discrete nonlinear problems using Newton/Picard method; -solution of the linear problems by iterative linear solvers (preconditioned conjugated gradient, preconditioned Krylov subspace methods, algebraic multigrid, and so on).…”

mentioning

confidence: 99%

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

2011

View full text Add to dashboard Cite

The need of accurate and efficient numerical schemes to solve Richards' equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards' equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated-unsaturated soils with hydraulic properties described by van Genuchten-Mualem models. The numerical results obtained are compared with those provided by the modified Picard-preconditioned conjugated gradient (Krylov subspace) approach.

show abstract

“…A variety of methods for numerical treatment of the Richards equation (or, in general, convection-diffusion-reaction systems) have been published in recent times, see e.g. Kees et al [7], Sembera et al [8], Solin and Kuraz [9], Tocci et al [11], Miller et al [12] and Kuraz et al [13,14].…”

mentioning

confidence: 99%