High-order Accurate Numerical Methods for Solving the Space FractionalAdvection-dispersion Equation

Journal of Applied & Computational Mathematics

article

Fenga L¹  Zhuangb P²  Liu F¹  Turnera I¹ 

[1] School of Mathematical Sciences, Queensland University of Technology;School of Mathematical Sciences, Xiamen University

关键词: Finite volume method;    Riemann-Liouville fractional derivative;    Fractional advection-dispersion equation;    Crank-Nicolson scheme;    Extrapolation method;   

DOI  :  10.4172/2168-9679.1000279

来源: Hilaris Publisher

【 摘 要 】

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.

【 授权许可】

Unknown