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

TL;DR

This paper introduces a second-order accurate implicit difference scheme for solving one- and two-dimensional time fractional reaction-diffusion equations with variable coefficients, proving stability, convergence, and demonstrating efficiency through numerical simulations.

Contribution

It develops a novel second-order implicit finite difference scheme for variable coefficient time fractional reaction-diffusion equations with rigorous stability and convergence analysis.

Findings

The scheme is unconditionally stable and converges with order O(τ^2 + h^2).

Numerical simulations confirm the scheme's high accuracy and efficiency.

Preconditioned iterative methods effectively solve the resulting linear systems.

Abstract

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.