Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes

Abstract

Finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and $L$1 approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. Stability of the proposed scheme is given using the discrete energy method. The numerical scheme is $\mathcal{O}$ ($N$^{−min{2−$α$,$rα$}}) accurate in time, where $α$ (0 < $α$ < 1) is the order of the fractional time derivative, $r$ is an index of the mesh partition, and it is second order accurate in space. Extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete $l$²-norm and the $l$^∞-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.