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

• * Corresponding author: Maria Alessandra Ragusa
• 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.

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

 Citation:

• 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 $0.10000$ $3.19826e-003$ - $0.05000$ $2.61740e-004$ $3.61108$ $0.02500$ $1.90572e-005$ $3.77973$ $0.01250$ $1.33477e-006$ $3.83567$ $0.00625$ $8.92357e-008$ $3.90283$

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 $0.10000$ $3.77425e-003$ - $0.05000$ $1.20417e-003$ $1.64815$ $0.02500$ $3.55418e-004$ $1.76045$ $0.01250$ $1.02360e-004$ $1.79586$ $0.00625$ $2.69907e-005$ $1.92312$

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 $0.10000\pi$ $3.26587e-004$ - $0.05000\pi$ $2.60038e-005$ $3.65067$ $0.02500\pi$ $1.81670e-006$ $3.83933$ $0.01250\pi$ $1.26448e-007$ $3.84471$ $0.00625\pi$ $8.13028e-009$ $3.95909$

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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