Compact Crank–Nicolson Schemes for a Class of Fractional Cattaneo Equation in Inhomogeneous Medium

In this paper, a compact Crank–Nicolson scheme is proposed and analyzed for a class of fractional Cattaneo equation. In developing the scheme, the Crank–Nicolson discretization is applied for the time derivatives both in classical and in fractional definitions. Moreover, a compact operator for the spatial derivative involving variable coefficient is derived. When the variable coefficient is replaced by a unit constant, it reveals a particularly significant situation that the derived compact operator degenerates to the common four-order compact operator for Laplacian. It is proved that the scheme is stable and convergent in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} semi-norm via energy method. The convergence orders are 3-γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 3-\gamma $$\end{document} in time and 4 in space, where γ∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (1,2)$$\end{document} is the order of fractional derivative. In addition, a compact Crank–Nicolson alternating direction implicit (ADI) scheme is constructed for the 2D case and the corresponding theoretical analysis is also presented. The derived ADI scheme combines the discretization operators for time both in classical and in fractional forms, which allows the utilization of the modified ADI scheme to reduce the storage requirements and the consumption of CPU time. The applicability and the accuracy of the scheme are demonstrated by numerical experiments in 1D and 2D cases.