Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative γ   lies in the interval 12,1. A correction term is added to maintain high accuracy when γ∈0,12. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.