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Upwind-Based Numerical Approximation of a Space-Time Fractional Advection-Dispersion Equation for Groundwater Transport Within Fractured Systems

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Fractional Derivatives with Mittag-Leffler Kernel

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

Abstract

Modelling groundwater transport in fractured aquifer systems is complex due to the uncertainty associated with delineating the specific fractures along which water and potential contaminants could be transported. The resulting uncertainty in modelled contaminant movement has implications for the protection of the environment, where inadequate mitigation or remediation measures could be employed. To improve the governing equation for groundwater transport modelling, the Atangana–Baleanu in Caputo sense (ABC) fractional derivative is applied to the advection-dispersion equation with a focus on the advection term to account for anomalous advection. Boundedness, existence and uniqueness for the developed advection-focused transport equation is presented. In addition, a semi-discretisation analysis is performed to demonstrate the equation stability in time. Augmented upwind schemes are investigated as they have been found to address stability problems when solute transport is advection-dominated. The upwind-based schemes are developed, and stability analysis conducted, to facilitate the solution of the complex equation. The numerical stability analysis found the upwind Crank–Nicolson to be the most stable, and is thus recommended for use with the ABC fractional advection-dispersion equation.

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Authors and Affiliations

  1. Faculty of Agricultural and Natural Sciences, Institute for Groundwater Studies, University of the Free State, Bloemfontein, 9301, Free State, South Africa

    A. Allwright & A. Atangana

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  1. A. Allwright
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  2. A. Atangana
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Correspondence to A. Atangana .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Allwright, A., Atangana, A. (2019). Upwind-Based Numerical Approximation of a Space-Time Fractional Advection-Dispersion Equation for Groundwater Transport Within Fractured Systems. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_18

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  • DOI: https://doi.org/10.1007/978-3-030-11662-0_18

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-11662-0

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