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## Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems

 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

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.

Citation: Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020025
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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 $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} v\frac{0.5 \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } +\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unconditionally stable / Conditionally stable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} \frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ 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 $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n}  Scheme Assumptions Stability condition Upwind (explicit)$ \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{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} \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}
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