Error analysis of a finite difference scheme on a modified graded mesh for a time-fractional diffusion equation

This paper is concerned with a finite difference scheme on a new modified graded mesh for a time fractional diffusion equation with a Caputo fractional derivative of order α∈(0,1). At first, the construction and some basic properties of this new modified graded mesh are investigated and on its basis the L1 scheme is applied to approximate the Caputo derivative. Meanwhile, the standard center finite difference scheme on a uniform mesh is used to discretize the diffusion term. Then, stability and convergence of the proposed scheme in the maximum norm are proved. The convergence result shows that on this modified graded mesh one attains an optimal 2−α rate for the L1 scheme. Finally, the presented theoretical results are supported by some numerical experiments.