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_\sigma $ 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 $ L^2 $-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 + \alpha) $-$ th $ order convergence. The numerical results show that the error estimates are sharp.
Citation: Jianxiong Cao, Yuexin Xing. Error analysis of a high-order compact ADI scheme for 2D time-fractional convection-diffusion equation with weakly singular solutions[J]. Electronic Research Archive, 2025, 33(8): 4763-4784. doi: 10.3934/era.2025214
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_\sigma $ 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 $ L^2 $-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 + \alpha) $-$ th $ order convergence. The numerical results show that the error estimates are sharp.
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Jianxiong Cao, Yuexin Xing. Error analysis of a high-order compact ADI scheme for 2D time-fractional convection-diffusion equation with weakly singular solutions[J]. Electronic Research Archive, 2025, 33(8): 4763-4784. doi: 10.3934/era.2025214
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