H1-norm error analysis of an ADI compact finite difference method for a two-dimensional time-fractional reaction-diffusion equation with variable coefficients

Abstract

This paper introduces a robust numerical approach based on an alternating implicit direction (ADI) compact finite difference scheme for approximating the solution of a variable coefficient time fractional reaction-diffusion (TFRD) model in two space dimensions. The model is characterized by initial weak singularity. We apply the L1 formula for discretization of the temporal fractional derivative (TFD) on a graded mesh while the space derivatives are approximated by a high-order ADI compact finite difference scheme. The solvability of this method is investigated. We present a framework for examining stability result and $H^1$-norm global error estimate of the proposed scheme. Numerical experiment is carried out to demonstrate the accuracy of the algorithm and to verify the theoretical results. We compare the computed results on the graded grids with those on the uniform grid to show the advantage of the graded grids method. The present study is the first work on design and analysis of L1-ADI method for the TFRD model with variable coefficients in two dimensions.