Numerical methods for the time-fractional diffusion equation: A review

This review paper focuses on the numerical solution of the time-fractional diffusion equation using various discretization techniques. For the time-fractional derivative, we consider methods such as L-type approximations and Grünwald-Letnikov-based formulas, while for the spatial diffusion term, we utilize the compact finite difference method, finite element method, spectral element method, meshless method, Chebyshev spectral method, and finite block method. In addition, stability and convergence theorems are presented, accompanied by numerical examples that confirm the theoretical results.