# American Institute of Mathematical Sciences

October  2021, 26(10): 5495-5508. doi: 10.3934/dcdsb.2020355

## 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  October 2021 Early access  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, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355
##### References:

show all references

##### 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 $0.10000$ $4.79240e-003$ - $0.05000$ $1.46420e-003$ $1.71063$ $0.02500$ $4.21902e-004$ $1.79513$ $0.01250$ $1.14998e-004$ $1.87530$ $0.00625$ $2.95945e-005$ $1.95821$
 [1] Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583 [2] Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051 [3] Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021153 [4] Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917 [5] Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029 [6] Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030 [7] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [8] Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873 [9] Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247 [10] Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037 [11] Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1173-1185. doi: 10.3934/jimo.2020016 [12] Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317 [13] Yan Gu, Nobuo Yamashita. Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 487-510. doi: 10.3934/naco.2020047 [14] Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078 [15] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037 [16] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040 [17] Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007 [18] Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035 [19] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [20] Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057

2020 Impact Factor: 1.327