Consistency of iterative operator‐splitting methods: Theory and applications

Geiser, Jürgen

doi:10.1002/num.20422

Cited by 15 publications

(8 citation statements)

References 19 publications

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“…[20] Although seldom recognized, this sequential flux treatment constitutes an "operator-splitting" (OS) approach, a numerical method popular for partial differential equations (PDEs) describing multiple spatially distributed processes with widely different time scales and behavior (e.g., see Geiser [2010] for general theory, Steefel and MacQuarrie [1996] for common geochemical applications, and Kavetski et al [2003] and Schoups et al [2010] for a discussion in rainfall-runoff modeling contexts). At the expense of additional "splitting" error, OS strategies allow the application of specialized numerical approximations to each individual process.…”

Section: Formulation Of Governing Model Equationsmentioning

confidence: 99%

Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes

Clark

Kavetski

2010

Water Resources Research

148

225

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[1] A major neglected weakness of many current hydrological models is the numerical method used to solve the governing model equations. This paper thoroughly evaluates several classes of time stepping schemes in terms of numerical reliability and computational efficiency in the context of conceptual hydrological modeling. Numerical experiments are carried out using 8 distinct time stepping algorithms and 6 different conceptual rainfallrunoff models, applied in a densely gauged experimental catchment, as well as in 12 basins with diverse physical and hydroclimatic characteristics. Results show that, over vast regions of the parameter space, the numerical errors of fixed-step explicit schemes commonly used in hydrology routinely dwarf the structural errors of the model conceptualization. This substantially degrades model predictions, but also, disturbingly, generates fortuitously adequate performance for parameter sets where numerical errors compensate for model structural errors. Simply running fixed-step explicit schemes with shorter time steps provides a poor balance between accuracy and efficiency: in some cases daily-step adaptive explicit schemes with moderate error tolerances achieved comparable or higher accuracy than 15 min fixed-step explicit approximations but were nearly 10 times more efficient. From the range of simple time stepping schemes investigated in this work, the fixed-step implicit Euler method and the adaptive explicit Heun method emerge as good practical choices for the majority of simulation scenarios. In combination with the companion paper, where impacts on model analysis, interpretation, and prediction are assessed, this two-part study vividly highlights the impact of numerical errors on critical performance aspects of conceptual hydrological models and provides practical guidelines for robust numerical implementation.Citation: Clark, M. P., and D. Kavetski (2010), Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes, Water Resour. Res., 46, W10510,

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Section: Formulation Of Governing Model Equationsmentioning

confidence: 99%

Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes

Clark

Kavetski

2010

Water Resources Research

148

225

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“…For i = 3, we have: Here an optimization is possible by assuming that commutators are equal or at least zero, see [2] and [8].…”

Section: Computation Of the Iterative Splitting Method: Closed Formulmentioning

confidence: 99%

“…On the other hand, they can be used to accelerate the iterative process of solving partial differential equations, see [21]. In the next step, the generalization of iterative splitting schemes to unbounded operators allows them be applied to partial differential equations, see [8], [9]. In this paper, we deal with a general scheme, a so called multi-stage scheme, which gives a significant improvement in terms of accuracy, numerical stability and reduction of local and global errors.…”

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confidence: 99%

An iterative splitting method via waveform relaxation

Geiser

2011

International Journal of Computer Mathematics

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“…The intermediate time‐level is given by t 0 = 0,…, t N = T . We obtain a Cauchy initial value problem, see [27].…”

Section: Analytical Solutions and Mass Computationmentioning

confidence: 99%

“…Furthermore, we could also embed semianalytical solutions of coupled convection–reaction problems that arise out of mobile and immobile models. Such coupling is done with iterative splitting methods, which combine the analytical parts of each equation with successive iterations, see [7].…”

Section: Introductionmentioning

confidence: 99%

Mobile and immobile gaseous transport: Embedded analytical solutions to finite volume methods

Geiser

2011

Numerical Methods Partial

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The motivation is driven by deposition processes based on chemical vapor problems. The underlying model problem is based on coupled transport–reaction equations with mobile and immobile areas. We deal with systems of ordinary and partial differential equations. Such equation systems are delicate to solve and we introduce a novel solver method, that takes into account ways to solve analytically parts of the transport and reaction equations. The main idea is to embed the analytical and semianalytical solutions, which can then be explicitly given to standard numerical schemes of higher order. The numerical scheme is based on flux‐based characteristic methods, which is a finite volume method. Such a method is an attractive alternative to the standard numerical schemes, which fully discretize the full equations. We instead reduce the computational time while embedding fast computable analytical parts. Here, we can accelerate the solver process, with a priori explicitly given solutions. We will focus on the derivation of the analytical solutions for general and special solutions of the characteristic methods that are embedded into a finite volume method. In the numerical examples, we illustrate the higher‐order method for different benchmark problems. Finally, the method is verified with realistic results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

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Consistency of iterative operator‐splitting methods: Theory and applications

Cited by 15 publications

References 19 publications

Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes

Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes

An iterative splitting method via waveform relaxation

Mobile and immobile gaseous transport: Embedded analytical solutions to finite volume methods

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