Comparison between iteration schemes for three-dimensional coordinate-transformed saturated–unsaturated flow model

An, Hyunuk; Ichikawa, Yutaka; Tachikawa, Yasuto; Shiiba, Michiharu

doi:10.1016/j.jhydrol.2012.08.056

Cited by 15 publications

(11 citation statements)

References 49 publications

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“…Indeed, the Picard algorithm exhibits slower convergence rates than the Newton algorithm in general. However, the Newton algorithm implies the computation and storage of the Jacobian matrix associated with the problem considered, which implies a huge increase in memory needs compared to the Picard Algorithm (see for example [59][60][61]). A Jacobian-free Newton algorithm does exist, but it still requires more memory than the Picard algorithm.…”

Section: Methodsmentioning

confidence: 99%

An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale

Orgogozo

Renon

Soulaine

et al. 2014

Computer Physics Communications

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⃝ Massively parallel computation Transfers in soils Porous media a b s t r a c tIn this paper we present a massively parallel open source solver for Richards equation, named the RichardsFOAM solver. This solver has been developed in the framework of the open source generalist computational fluid dynamics tool box OpenFOAM R ⃝ and is capable to deal with large scale problems in both space and time. The source code for RichardsFOAM may be downloaded from the CPC program library website.It exhibits good parallel performances (up to ∼90% parallel efficiency with 1024 processors both in strong and weak scaling), and the conditions required for obtaining such performances are analysed and discussed. These performances enable the mechanistic modelling of water fluxes at the scale of experimental watersheds (up to few square kilometres of surface area), and on time scales of decades to a century. Such a solver can be useful in various applications, such as environmental engineering for long term transport of pollutants in soils, water engineering for assessing the impact of land settlement on water resources, or in the study of weathering processes on the watersheds. Program summaryProgram title: RichardsFOAM Catalogue identifier: AEUF_v1_0 Program summary URL: L. Orgogozo et al. / Computer Physics Communications ( ) -External routines: OpenFOAM R ⃝ (version 2.0.1 or later) Nature of problem:This software solves the non-linear three-dimensional transient Richards equation, which is a very popular model for water transfer in variably saturated porous media (e.g.: soils). It is designed to take advantage of the massively parallel computing performance of OpenFOAM R ⃝ . The goal is to be able to model natural hydrosystems on large temporal and spatial scales. Solution method:A mixed implicit (FVM in the object oriented OpenFOAM R ⃝ framework) and explicit (FVC in the object oriented OpenFOAM R ⃝ framework) discretization of the equation with a backward time scheme is coupled with a linearization method (Picard algorithm). Due to the linearization loop the final solution of each time step tends towards a fully implicit solution. The implementation has been carried out with a concern for robustness and parallel efficiency. Restrictions:The choice of the maximum and initial time steps must be made carefully in order to avoid stability problems. A careful convergence study of mesh cell size, linear solver precision and linearization method precision must be undertaken for each considered problem, depending on the precision required for the expected results, the spatial and temporal scales at stake, and so on. Finally, the solver in its current version only handles meshes with a constant cell volume (a crash will not necessarily occur with an irregular mesh but some problems may arise with the convergence criterion of the linearization method). Running time:Highly variable, depending on the mesh size and the number and nature of cores involved. The test run provided requires less than 2 s on a 64 bit machine with Intel R ⃝ CoreT...

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Section: Methodsmentioning

confidence: 99%

An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale

Orgogozo

Renon

Soulaine

et al. 2014

Computer Physics Communications

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“…The first example was presented by Haverkamp et al 13 , where boundary condition was -61.5 cm for downstream and -20.7 cm for upstream and initial condition was -61.5 cm. They presented eqs (29) and (30) for determining the relationship between K(h) and (h) The second example given by Van Genuchten 14 was an unsaturated, homogeneous soil column. Downstream boundary condition for this soil was -1000 cm, and upstream boundary condition was -75 cm.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The aim of their study was to consider vertical columns of infiltration coupled with lateral transfer of mass through the boundary conditions. Different numerical models have been proposed for modelling water flow in unsaturated-saturated zones [18][19][20][21][22][23][24][25][26][27][28][29] . Here we use differential quadrature method (DQM) for solving problems of water flow in unsaturated zones.…”

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confidence: 99%

Numerical Solution for One-Dimensional Richards' Equation Using Differential Quadrature Method

Nikzad¹,

Eslamian²,

Soleymannejad³

et al. 2016

Current Science

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Nonlinear nature of Richards' equation has attracted attention for providing analytical and numerical solution for this equation. In the present study, differential quadrature method (DQM) is used for presenting a numerical solution for one-dimensional Richards' equation in mixed and h-based forms, where pressure head is the dependent variable. Our results show that DQM requires less computational effort compared to finite difference technique, and can be applied for solving other similar nonlinear equations.

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“…These techniques include Finite-Di erence Method (FDM), Finite-Element Method (FEM), Finite Volume Method (FVM), hp-FEM and time splitting method [2][3][4][5][6][7][8][9][10][11]. Analytical solutions, on the other hand, are mainly o ered for one-dimensional ow of water through the soil and for restrictive boundary and initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

2017

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Abstract. Fluid in ltration into unsaturated soil is of vital signi cance from many perspectives. Mathematically, such transient in ltrations are described by Richards' equation: a nonlinear parabolic Partial Di erential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1D horizontal and vertical water in ltration are derived and compared to numerical nite-di erence method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water in ltration are derived for constant, no-ow, and sinusoidal boundary and initial conditions. The presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use di erent computational techniques.

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Comparison between iteration schemes for three-dimensional coordinate-transformed saturated–unsaturated flow model

Cited by 15 publications

References 49 publications

An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale

An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale

Numerical Solution for One-Dimensional Richards' Equation Using Differential Quadrature Method

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

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