A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity

Abstract

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete $$L^{2}$$ -norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.