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An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formulais presented for a class of one-dimensional time fractional reaction-diffusionequations with variable coefficients and time drift term. The unconditionalstability and convergence of this scheme are proved rigorously by the discreteenergy method, and the optimal convergence order in the $L_2$-norm is$\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Then, thesame measure is exploited to solve the two-dimensional case of this problem anda rigorous theoretical analysis of the stability and convergence is carriedout. Several numerical simulations are provided to show the efficiency andaccuracy of our proposed schemes and in the last numerical experiment of thiswork, three preconditioned iterative methods are employed for solving thelinear system of the two-dimensional case.

A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term