Error analysis of a high-order compact ADI scheme for 2D time-fractional convection-diffusion equation with weakly singular solutions

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Abstract

In this paper, we propose a high-order numerical method for the 2D time-fractional convection-diffusion equation with weakly singular solutions, where the time Caputo fractional derivative is approximated by using the L2-1_σ formula on a nonuniform graded mesh, and the space derivatives are discretized by a fourth-order compact finite difference scheme on a uniform mesh. The fully discrete compact ADI scheme is established by adding a high-order term. The stability and the convergence analyses of the scheme are analyzed in L²-norm by using the discrete energy method. It has been proved that the introduced numerical scheme has spatial fourth-order convergence, and temporal optimal (1 + α)-th order convergence. The numerical results show that the error estimates are sharp.

Keywords

time-fractional convection-diffusion equation / L2-1 _σ formula / ADI method / stability / convergence

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