A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term
This work provides a rigorous theoretical analysis and efficient numerical scheme for solving a class of fractional reaction-diffusion equations, which is incremental as it extends existing L2-1σ methods to include variable coefficients and a time drift term.
An implicit finite difference scheme based on the L2-1σ formula is developed for time fractional reaction-diffusion equations with variable coefficients and time drift term, achieving unconditional stability and optimal convergence order O(τ² + h²) in the L₂-norm, as demonstrated by numerical experiments.
An implicit finite difference scheme based on the $L2$-$1_σ$ 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 $\mathcal{O}(τ^2 + h^2)$ with time step $τ$ 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.