A mass balance based numerical method for the fractional advection-dispersion equation: theory and application

Xiaoxian Zhang, John W. Crawford, Lynda K. Deeks, Marc I. Stutter, A. Glyn Bengough, Iain M. Young

Research output: Contribution to journalArticle

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Abstract

The inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement, and one approach to model solute transport through such media is the continuous time random walk (CTRW). One special asymptotic case of the CTRW is the fractional advection-dispersion equation (FADE), which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water. Most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically, but the numerical solution of the FADE is not trivial. In this paper we present a finite volume approach to solve the FADE where the spatial derivative of the dispersion term is fractional. We also give methods to solve different boundary conditions often encountered in practical applications. The linear system resulting from the temporal-spatial discretization is solved using a semi-implicit scheme. The numerical method is derived on the basis of mass balance, and its accuracy is tested against analytical solutions. The method is then applied to simulate tracer movement in a stream and a near-saturated hillslope in a naturally structured upland podzol field in northeast Scotland.
Original languageEnglish
Article numberW07029
Number of pages10
JournalWater Resources Research
Volume41
Issue number7
DOIs
StatePublished - Jul 2005

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advection
numerical method
mass balance
hydrology
boundary condition
water
Podzol
solute transport
hillslope
tracer

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Zhang, X., Crawford, J. W., Deeks, L. K., Stutter, M. I., Bengough, A. G., & Young, I. M. (2005). A mass balance based numerical method for the fractional advection-dispersion equation: theory and application. Water Resources Research, 41(7), [W07029]. DOI: 10.1029/2004WR003818

Zhang, Xiaoxian; Crawford, John W.; Deeks, Lynda K.; Stutter, Marc I.; Bengough, A. Glyn; Young, Iain M. / A mass balance based numerical method for the fractional advection-dispersion equation : theory and application.

In: Water Resources Research, Vol. 41, No. 7, W07029, 07.2005.

Research output: Contribution to journalArticle

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Zhang, X, Crawford, JW, Deeks, LK, Stutter, MI, Bengough, AG & Young, IM 2005, 'A mass balance based numerical method for the fractional advection-dispersion equation: theory and application' Water Resources Research, vol 41, no. 7, W07029. DOI: 10.1029/2004WR003818

A mass balance based numerical method for the fractional advection-dispersion equation : theory and application. / Zhang, Xiaoxian; Crawford, John W.; Deeks, Lynda K.; Stutter, Marc I.; Bengough, A. Glyn; Young, Iain M.

In: Water Resources Research, Vol. 41, No. 7, W07029, 07.2005.

Research output: Contribution to journalArticle

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Zhang X, Crawford JW, Deeks LK, Stutter MI, Bengough AG, Young IM. A mass balance based numerical method for the fractional advection-dispersion equation: theory and application. Water Resources Research. 2005 Jul;41(7). W07029. Available from, DOI: 10.1029/2004WR003818