# American Institute of Mathematical Sciences

## 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

Received  June 2020 Revised  September 2020 Published  December 2020

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.

Citation: Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020355
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##### References:
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 $0.10000$ $3.19826e-003$ - $0.05000$ $2.61740e-004$ $3.61108$ $0.02500$ $1.90572e-005$ $3.77973$ $0.01250$ $1.33477e-006$ $3.83567$ $0.00625$ $8.92357e-008$ $3.90283$
 Maximum Estimated $\mathit{h}_{x}=\mathit{h}_{y}$ Absolute Error Convergence Rate $0.10000$ $3.19826e-003$ - $0.05000$ $2.61740e-004$ $3.61108$ $0.02500$ $1.90572e-005$ $3.77973$ $0.01250$ $1.33477e-006$ $3.83567$ $0.00625$ $8.92357e-008$ $3.90283$
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 $0.10000$ $3.77425e-003$ - $0.05000$ $1.20417e-003$ $1.64815$ $0.02500$ $3.55418e-004$ $1.76045$ $0.01250$ $1.02360e-004$ $1.79586$ $0.00625$ $2.69907e-005$ $1.92312$
 Maximum Estimated $\mathit{k}_{t}$ Absolute Error Convergence Rate $0.10000$ $3.77425e-003$ - $0.05000$ $1.20417e-003$ $1.64815$ $0.02500$ $3.55418e-004$ $1.76045$ $0.01250$ $1.02360e-004$ $1.79586$ $0.00625$ $2.69907e-005$ $1.92312$
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 $0.10000\pi$ $3.26587e-004$ - $0.05000\pi$ $2.60038e-005$ $3.65067$ $0.02500\pi$ $1.81670e-006$ $3.83933$ $0.01250\pi$ $1.26448e-007$ $3.84471$ $0.00625\pi$ $8.13028e-009$ $3.95909$
 Maximum Estimated $\mathit{h}_{x}=\mathit{h}_{y}$ Absolute Error Convergence Rate $0.10000\pi$ $3.26587e-004$ - $0.05000\pi$ $2.60038e-005$ $3.65067$ $0.02500\pi$ $1.81670e-006$ $3.83933$ $0.01250\pi$ $1.26448e-007$ $3.84471$ $0.00625\pi$ $8.13028e-009$ $3.95909$
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 $0.10000$ $4.79240e-003$ - $0.05000$ $1.46420e-003$ $1.71063$ $0.02500$ $4.21902e-004$ $1.79513$ $0.01250$ $1.14998e-004$ $1.87530$ $0.00625$ $2.95945e-005$ $1.95821$
 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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