High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

Abdollah Borhanifar; Maria Alessandra Ragusa; Sohrab Valizadeh

doi:10.3934/dcdsb.2020355

Article Contents

2021, Volume 26, Issue 10: 5495-5508. Doi: 10.3934/dcdsb.2020355

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High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

1.
Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran
2.
Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6-95125 Catania, Italy
3.
RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

^* Corresponding author: Maria Alessandra Ragusa
^* Corresponding author: Maria Alessandra Ragusa

Received: June 2020

Revised: September 2020

Early access: November 2020

Published: October 2021

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Abstract

In this paper, by combining of fractional centered difference approach with alternating direction implicit method, we introduce a mixed difference method for solving two-dimensional Riesz space fractional advection-dispersion equation. The proposed method is a fourth order centered difference operator in spatial directions and second order Crank-Nicolson method in temporal direction. By reviewing the consistency and stability of the method, the convergence of the proposed method is achieved. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed technique.

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Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35R11.

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Table 1. The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved spatial step sizes and $ \mathit{k}_{t} = 0.001 $

	Maximum	Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $	Absolute Error	Convergence Rate
$ 0.10000 $	$ 3.19826e-003 $	-
$ 0.05000 $	$ 2.61740e-004 $	$ 3.61108 $
$ 0.02500 $	$ 1.90572e-005 $	$ 3.77973 $
$ 0.01250 $	$ 1.33477e-006 $	$ 3.83567 $
$ 0.00625 $	$ 8.92357e-008 $	$ 3.90283 $

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Table 2. The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved temporal step sizes and $ \mathit{h}_{x} = \mathit{h}_{y} = 0.001 $

	Maximum	Estimated
$ \mathit{k}_{t} $	Absolute Error	Convergence Rate
$ 0.10000 $	$ 3.77425e-003 $	-
$ 0.05000 $	$ 1.20417e-003 $	$ 1.64815 $
$ 0.02500 $	$ 3.55418e-004 $	$ 1.76045 $
$ 0.01250 $	$ 1.02360e-004 $	$ 1.79586 $
$ 0.00625 $	$ 2.69907e-005 $	$ 1.92312 $

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Table 3. The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved spatial step sizes and $ \mathit{k}_{t} = 0.001 $

	Maximum	Estimated
$ \mathit{h}_{x}=\mathit{h}_{y} $	Absolute Error	Convergence Rate
$ 0.10000\pi $	$ 3.26587e-004 $	-
$ 0.05000\pi $	$ 2.60038e-005 $	$ 3.65067 $
$ 0.02500\pi $	$ 1.81670e-006 $	$ 3.83933 $
$ 0.01250\pi $	$ 1.26448e-007 $	$ 3.84471 $
$ 0.00625\pi $	$ 8.13028e-009 $	$ 3.95909 $

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Table 4. The maximum errors and convergence rates for the modified Crank-Nicolson ADI method for solving two dimensional RSFADE with halved temporal step sizes and $ \mathit{h}_{x} = \mathit{h}_{y} = 0.001\pi $

	Maximum	Estimated
$ \mathit{k}_{t} $	Absolute Error	Convergence Rate
$ 0.10000 $	$ 4.79240e-003 $	-
$ 0.05000 $	$ 1.46420e-003 $	$ 1.71063 $
$ 0.02500 $	$ 4.21902e-004 $	$ 1.79513 $
$ 0.01250 $	$ 1.14998e-004 $	$ 1.87530 $
$ 0.00625 $	$ 2.95945e-005 $	$ 1.95821 $

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