  An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula
is presented for a class of one-dimensional time fractional reaction-diffusion
equations with variable coefficients and time drift term. The unconditional
stability and convergence of this scheme are proved rigorously by the discrete
energy 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, the
same measure is exploited to solve the two-dimensional case of this problem and
a rigorous theoretical analysis of the stability and convergence is carried
out. Several numerical simulations are provided to show the efficiency and
accuracy of our proposed schemes and in the last numerical experiment of this
work, three preconditioned iterative methods are employed for solving the
linear system of the two-dimensional case.
