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

Amy Allwright; Abdon Atangana

doi:10.3934/dcdss.2020025

Article Contents

2020, Volume 13, Issue 3: 443-466. Doi: 10.3934/dcdss.2020025

This issue Previous Article A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel Next Article Models of fluid flowing in non-conventional media: New numerical analysis

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
^* Corresponding author: A. Allwright

Received: May 31, 2018

Revised: June 30, 2018

Published: December 2019

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Abstract

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.

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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
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}} $	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) $
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}} $	Conditionally stable $ \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) $
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}} $	Conditionally stable $ \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} <\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $ $ \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{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
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}} $	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) $
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}} $	$ \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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