Implementation of Richardson extrapolation in an efficient adaptive time stepping method: applications to reactive transport and unsaturated flow in porous media

Belfort, Benjamin; Carrayrou, Jérôme; Lehmann, François

doi:10.1007/s11242-006-9090-3

Cited by 16 publications

(8 citation statements)

References 29 publications

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“…A priori estimation methods, such as those presented by Schneid et al (2004) and Bause and Merz (2005), attempt to determine a bound on the truncation error before the computation of the solution. However, as Belfort et al (2007) state, this can be difficult to calculate with sufficient accuracy and depends on the convergence rate and derivates of the function, which can be specific to the numerical scheme and the problem.…”

Section: Time-discretization In Regional Groundwater Modelsmentioning

confidence: 99%

Simple Automatic Time‐Stepping for Improved Simulation of Groundwater Hydrographs

Jackson

2011

Groundwater

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An automatic time-stepping algorithm is presented, based on the intensity of driving groundwater recharge, that improves the simulation of groundwater level fluctuations in regional models while maintaining model run-times. The algorithm is implemented in the ZOOMQ3D finite difference groundwater flow code and controls the discretization of time using two user-defined criteria: a maximum time-step length and a maximum recharge per time-step. Daily recharge is accumulated in time until either of these criteria is violated when the model then calculates a solution. The efficiency and accuracy of the algorithm is tested using an idealized groundwater model and an existing regional groundwater model of a UK aquifer. The approach is illustrated using simulations of high groundwater levels and associated groundwater flood events in a responsive, high-diffusivity aquifer. Simulations using the automatic time-stepping technique are presented that reduce the maximum absolute error in groundwater level by 45% and the run-time by 51% compared to models using conventional, a priori defined stress-periods and time-steps.

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Section: Time-discretization In Regional Groundwater Modelsmentioning

confidence: 99%

Simple Automatic Time‐Stepping for Improved Simulation of Groundwater Hydrographs

Jackson

2011

Groundwater

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“…To allow time step management with SNIA during the simulation, we use the Richardson extrapolation which is an adaptive time stepping procedure based on the posteriori error estimate (Belfort et al 2007). First, a single computation is performed using a time step of Δt.…”

Section: The Sequential Non-iterative Approach (Snia)mentioning

confidence: 99%

On the Efficiency of the Direct Substitution Approach for Reactive Transport Problems in Porous Media

Fahs

Carrayrou

Younès

et al. 2008

Water Air Soil Pollut

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Nonlinear reactive transport problems can be solved using the Operator Splitting (OS) approach, where transport and reaction processes are separated or the Direct Substitution Approach (DSA) where chemical and transport equations are solved simultaneously. The OS techniques can be very attractive, but are known to introduce splitting errors with SNIA (Non Iterative OS) and have low convergence rate with SIA (Iterative OS). These problems are avoided with DSA which is more robust than OS schemes. On the other hand, DSA is more complicated and very demanding in terms of computing time and memory requirements. This can make DSA less efficient than OS schemes especially for fine discretizations and chemically simple problems. In this work, DSA, SIA and SNIA are combined with a new sparse direct (unifrontal/multifrontal) solver. The efficiency of this solver is not dependent on the matrix conditioning. The performance of the three approaches is studied for two transport problems with simple and difficult chemical reactions and for different number of unknowns. Results show that when combined with an efficient sparse direct solver, DSA is more efficient than SIA and SNIA even for chemically simple problems and large number of unknowns.

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“…Nevertheless, the EBATS method has been shown to be quite simple and robust [78,83] which explains its popularity [27,28,30,39]. We have also used this simple method, however the additional use of a multistep BDF method or the Richardson method may lead to more efficient solving procedure (truncation error control, optimality of the sequence of time steps; see for example [79,75,81,82]). …”

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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Implementation of Richardson extrapolation in an efficient adaptive time stepping method: applications to reactive transport and unsaturated flow in porous media

Cited by 16 publications

References 29 publications

Simple Automatic Time‐Stepping for Improved Simulation of Groundwater Hydrographs

Simple Automatic Time‐Stepping for Improved Simulation of Groundwater Hydrographs

On the Efficiency of the Direct Substitution Approach for Reactive Transport Problems in Porous Media

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

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