High-order algorithm for the two-dimension Riesz space-fractional diffusion equation

ABSTRACT

In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order $\mathcal{O}({\tau }^2+h_x^2+h_y^2)$, where τ, $h_x$ and $h_y$ are the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein.

KEYWORDS:

• Riesz derivative
• second-order numerical approximation formula
• Crank–Nicolson technique

CLASSIFICATION:

• 65A05
• 65D15
• 65D25
• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Hengfei Ding http://orcid.org/0000-0003-4044-6499

Additional information

Funding

The work was partially supported by the National Natural Science Foundation of China under Grant No. [11561060], the Scientific Research Program for Young Teachers of Tianshui Normal University under Grant No. [TSA1405], and Tianshui Normal University Key Construction Subject Project (Big data processing in dynamic image).