High-order Accurate Numerical Methods for Solving the Space Fractional Advection-dispersion EquationFenga L1, Zhuangb P2, Liu F1* and Turnera I1
- *Corresponding Author:
- Liu F
School of Mathematical Sciences
Queensland University of Technology
GPO Box 2434, Brisbane, Qld 4001, Australia
E-mail: [email protected]
Received date: November 14, 2015; Accepted date: December 14, 2015; Published date: December 18, 2015
Citation: Fenga L, Zhuangb P, Liu F, Turnera I (2015) High-order Accurate Numerical Methods for Solving the Space Fractional Advection-dispersion Equation. J Appl Computat Math 4:279. doi:10.4172/2168-9679.1000279
Copyright: © 2015 Fenga L, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
In this paper, we consider a type of space fractional advection-dispersion equation, which is obtained from the classical advection-diffusion equation by replacing the spatial derivatives with a generalized derivative of fractional order. Firstly, we utilize the modified weighted and shifted Grunwald difference operators to approximate the Riemann-Liouville fractional derivatives and present the finite volume method. 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). Furthermore, we apply an extrapolation method to improve the convergence order, which can be O(τ4 + h4). Finally, two numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis.