Stability and convergence of a finite volume method for the space fractional advection–dispersion equation

Hejazi, Hala; Moroney, Timothy J.; Liu, F.

doi:10.1016/j.cam.2013.06.039

Cited by 123 publications

(67 citation statements)

References 25 publications

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“…Previously [24] we have shown this discretisation to be of first order spatial accuracy for the constant coefficient case. For the advective flux V ( )C ( ) we use a standard averaging scheme…”

Section: Definition 5 (Shifted Grünwald Formulas Onmentioning

confidence: 85%

“…Previously we have considered a finite volume method for the two-sided space-fractional advection-dispersion equation with constant coefficients [24], based on the Grünwald-Letnikov definition, where we proved the stability and convergence of the method. In this paper, we extend the method to solve the two-sided time-space fractional advection-dispersion equation with variable coefficients.…”

Section: Definition 2 (Riemann-liouville Fractional Derivatives On [ mentioning

confidence: 93%

“…In the present method, the fractional shift = 1/2 is used, allowing us to build approximations of fractional derivatives at control volume faces ±1/2 in terms of function values at the nodes [24]. Hence we derive the fractionallyshifted Grünwald formulas…”

Section: Definition 5 (Shifted Grünwald Formulas Onmentioning

confidence: 99%

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A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

2013

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Abstract:We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach. PACS

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Section: Definition 5 (Shifted Grünwald Formulas Onmentioning

confidence: 85%

Section: Definition 2 (Riemann-liouville Fractional Derivatives On [ mentioning

confidence: 93%

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A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

2013

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“…A new generation of numerical algorithm belongs to the Finite Volume Method (FVM) [Hunter, B. et al 2016, Shewchuk, J.R. 2002, & Hejazi, H. et al 2014 . Its processes are that divide the solving area into a series of discrete non-overlapping control volume, and make sure each node has a control volume around; integral the differential equation to be solved at each little control volume to derive a set of discrete equations.…”

Section: Instructionmentioning

confidence: 99%

FVM for Solving Three Phase Percolation Model of Unsaturated-saturated Zone

Sun¹,

Shao²,

Cui³

2017

dtmse

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“…During the past decades, a plenty full of numerical schemes have been developed. These methods include finite difference approximation methods [6,7,8,9,10,11,12], finite volume methods [13,14], variational iteration method [15,16], spline functions [17,18], Tau method [19], generalized differential transform methods [20,21], the Adomian decomposition methods [22,23,24] and Chebyshev spectral method [25].…”

Section: Introductionmentioning

confidence: 99%

Mittag-Leffler-Pade approximations for the numerical solution of space and time fractional diffusion equations

Borhanifar

Valizadeh

2015

IJAMR

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original work is properly cited. AbstractAnomalous diffusion and non-exponential relaxation patterns can be described by a space -time fractional diffusion equation. This paper aims to present a Padé approximation for Mittag-Leffler function mixed finite difference method to develop a numerical method to obtain an approximate solution for the space and time fractional diffusion equation. The truncation error of the method is theoretically analyzed. It is proved that the numerical proposed method is unconditionally stable from the matrix analysis point of view. Finally, some numerical results are given, which demonstrate the efficiency of the approximate scheme.

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Stability and convergence of a finite volume method for the space fractional advection–dispersion equation

Cited by 123 publications

References 25 publications

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

FVM for Solving Three Phase Percolation Model of Unsaturated-saturated Zone

Mittag-Leffler-Pade approximations for the numerical solution of space and time fractional diffusion equations

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