A block-centered finite-difference method for the time-fractional diffusion equation on nonuniform grids

ABSTRACT

In this article, a block-centered finite-difference scheme is introduced to solve the time-fractional diffusion equation with a Caputo derivative of order α ∈ (0, 1) on nonuniform grids. The resulting scheme is second-order-accurate in space and (2 − α)-order-accurate in time, and the unconditional stability and convergence are proved theoretically. Moreover, numerical solutions of the unknown variable along with its first derivatives are obtained. Finally, numerical experiments, including boundary-layer and high-gradient problems, are carried out to support our theoretical analysis and indicate the efficiency of this method.

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions, which helped us to improve the results of this article. The authors also thank Dr. Dongwei Gui (Cele National Station of Observation & Research for Desert-Grassland Ecosystem in Xinjiang) for his support and encouragement in this work.