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

Research paper by L.B. Feng, P. Zhuang, F. Liu, I. Turner, J. Li

Indexed on: 18 Mar '16Published on: 19 Feb '16Published in: Computers & Mathematics with Applications



Abstract

In this paper, we propose high-order numerical methods for the Riesz space fractional advection–dispersion equations (RSFADE) on a finite 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 α∈(0,1)α∈(0,1) and β∈(1,2]β∈(1,2], respectively. Firstly, we utilize the weighted and shifted Grünwald difference operators to approximate the Riesz fractional derivative and present the finite 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 O(τ2+h2)O(τ2+h2). Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be O(τ4+h4)O(τ4+h4). Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent with theoretical analysis.

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