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This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0 , 1). A (3 − α) th-order numerical formula (called the L 2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of (3 − α) th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α ∈ (0 , 1). Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results. [ABSTRACT FROM AUTHOR]

In a recent paper, Ren and Liu proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients. Based on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods. [ABSTRACT FROM AUTHOR]