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

Juncu, Gheorghe; Nicola, Aurelian; Popa, Constantin

doi:10.1007/s11242-011-9831-9

Cited by 6 publications

(7 citation statements)

References 47 publications

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“…The computed results are depicted in Figure 4 which demonstrates that the process of infiltration can continue if there is a pool of water at ground surface maintained holding the pressure head for additional water at the soil surface. It is found that the best accuracy of the proposed method can reach up to 13 …”

Section: Steady-state Modeling Of Two-dimensional Subsurface Flow In mentioning

confidence: 99%

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Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method

Liu

Xiao

et al. 2017

Water

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Abstract:In this paper, a novel meshless method for the transient modeling of subsurface flow in unsaturated soils was developed. A linearization process for the nonlinear Richards equation using the Gardner exponential model to analyze the transient flow in the unsaturated zone was adopted. For the transient modeling, we proposed a pioneering work using the collocation Trefftz method and utilized the coordinate system in Minkowski spacetime instead of that in the original Euclidean space. The initial value problem for transient modeling of subsurface flow in unsaturated soils can then be transformed into the inverse boundary value problem. A numerical solution obtained in the spacetime coordinate system was approximated by superpositioning Trefftz basis functions satisfying the governing equation for boundary collocation points on partial problem domain boundary in the spacetime coordinate system. As a result, the transient problems can be solved without using the traditional time-marching scheme. The validity of the proposed method is established for several test problems. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for the transient modeling of subsurface flow in unsaturated soils.

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Section: Steady-state Modeling Of Two-dimensional Subsurface Flow In mentioning

confidence: 99%

“…Nonlinear physical relationships can be described using soil-water characteristic curves [6][7][8]. Since the Richards equation is highly nonlinear and cannot directly provide an analytical solution, modeling flow process in unsaturated soils is usually based on the numerical solutions of the Richards equation [9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method

Liu

Xiao

et al. 2017

Water

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“…There have been efforts to develop or apply robust, scalable preconditioners for Richards' equation (Jenkins et al, 2001;Jones and Woodward, 2001), and high-quality software packages like PETSc provide access to a number of modern preconditioners as well as linear and nonlinear solvers (Balay et al, 1997). However, uniformly robust preconditioning techniques and nonlinear solver performance for Richards' equation for remains an open research topic (Juncu et al, 2011;Lipnikov et al, 2016).…”

Section: Linear and Nonlinear Solversmentioning

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.…”

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confidence: 99%

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

Farthing

Ogden

2017

Soil Science Soc of Amer J

245

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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“…Also, there are a number of effective solution approaches based on Newton's method, see for e.g. [42,43,44]. These methods exhibit a quadratic convergence rate but are very sensitive to initial solution approximations.…”

Section: Cell-centered Multigridmentioning

confidence: 99%

A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

et al. 2019

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We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that are modeled using the Richards' equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards' equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards' equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared to the original MC estimator.

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

Cited by 6 publications

References 47 publications

Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method

Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method

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

A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

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