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March 2020, 13(3): 443-466. doi: 10.3934/dcdss.2020025
Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems
Amy Allwright , and Abdon Atangana
Institute for Groundwater Studies, Faculty of Agricultural and Natural Sciences, University of the Free State, 9301, Bloemfontein, Free State, South Africa
* Corresponding author: A. Allwright
Received June 2018 Revised July 2018 Published March 2019
Table(1)
The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.
Keywords: Fractional advection-dispersion equation, anomalous diffusion, anomalous advection, fractured groundwater transport, upwind numerical scheme, finite difference.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 443-466. doi: 10.3934/dcdss.2020025
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Table 1. Summary of the established stability condition, and corresponding assumption, for each numerical approximation scheme
Scheme Assumptions Stability condition
Upwind (explicit) $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } > \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Unstable
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Conditionally stable
$ \frac{4D_{L}}{ \left( \Delta x \right) ^{2}} +v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m}+\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <\frac{2 \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $
Upwind (implicit) $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Unconditionally stable / Conditionally stable
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2 \delta _{n,m}^{ \alpha } + \left( 2-2cos \phi \right) \beta _{m} \right) $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m} +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
Upwind CrankNicolson $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }>v\frac{0.5 \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } +\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Unconditionally stable / Conditionally stable
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( \delta _{n,m}^{ \alpha } + \beta _{m} \left( 1-cos \phi \right) \right) $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } <v\frac{0.5 \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }+\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Conditionally stable
$ \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} <\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $
$ \frac{4D_{L}}{ \left( \Delta x \right) ^{2}}+\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <\frac{2 \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 1-cos \phi \right) \beta _{m} $
Weighted upwinddownwind (explicit) $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } + v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Unstable
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } + v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Conditionally stable / Unstable
Weighted upwinddownwind (implicit) $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } +v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ Unconditionally stable / conditionally stable
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( \theta +1 \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } +v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $ $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) < 2 v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( 1- \theta \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
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Amy Allwright Abdon Atangana