A New L2 Type Difference Scheme for the Time-Fractional Diffusion Equation

In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference method, yielding a fully discrete system. We then establish the H1-norm stability and convergence of the time-stepping scheme on uniform meshes for the TFDE. In particular, we prove that the proposed scheme has (3−α)th-order accuracy, where α (0<α<1) is the order of the time-fractional derivative. Finally, numerical experiments for several test problems are carried out to validate the obtained theoretical results.