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arXiv · 2003.13923

High-order numerical methods for the Riesz space fractional advection-dispersion equations

Libo Feng·Pinghui Zhuang·Fawang Liu·Ian Turner·Jing Li
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Abstract

In this paper, we propose high-order numerical methods for the Riesz space fractional advection-dispersion equations (RSFADE) on a {f}inite domain. The RSFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order $\alpha\in(0,1)$ and $\beta\in(1,2]$, respectively. Firstly, we utilize the weighted and shifted Gr\"unwald difference operators to approximate the Riesz fractional derivative and present the {f}inite difference method for the RSFADE. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of $\mathcal {O}(\tau^2+h^2)$. Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be $\mathcal {O}(\tau^4+h^4)$. Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are excellent with the theoretical analysis.

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Libo Feng, Pinghui Zhuang, Fawang Liu, Ian Turner, Jing Li. 2020-03-31. High-order numerical methods for the Riesz space fractional advection-dispersion equations. https://arxiv.org/abs/2003.13923

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