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

In this article, a block-centered finite-difference scheme is introduced to solve the time-fractional diffusion equation with a Caputo derivative of order alpha is an element of (0, 1) on nonuniform grids. The resulting scheme is second-order-accurate in space and (2-alpha)-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.