A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation

This paper presents a high-order computational scheme for numerical solution of a time-fractional diffusion equation (TFDE). This scheme is discretized in time by means of L1-scheme and discretized in space using a compact finite difference method. Stability analysis of the method is discussed. Further, convergence analysis of the present numerical scheme is established and we show that this scheme is of O(Δt2-α+Δx4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({\Delta t}^{2-\alpha }+{\Delta x}^{4})$$\end{document} convergence, where α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} is the order of fractional derivative (FD) appearing in the governing equation and Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} and Δx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} are the step sizes in temporal and spatial direction, respectively. Three numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by finite element method and B-spline method. Comparison reveals that the proposed method is fast convergent and highly accurate. Moreover, the effect of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} on the numerical solution of TFDE is investigated. The CPU time of the present method is provided.