Optimal error estimates of the ADI scheme for the two-dimensional time-fractional cable equation

In this paper, a fully discrete alternating direction implicit (ADI) numerical method is developed for the two-dimensional (2-D) time-fractional Cable equation (TFCE) with weakly singular solutions on a non-uniform temporal grid, and the optimal error estimate is derived. In the proposed ADI method, the time-fractional Riemann-Liouville (R-L) derivative is approximated by the variable-step \(L1_R\) formula, and the spatial derivative is discretized by the second-order central difference scheme. Combined with the corresponding ADI algorithms, the overall computational cost is reduced significantly. Moreover, the theoretical stability and convergence analysis of the scheme are discussed in detail by means of the energy analysis method. Finally, numerical results show that the proposed method is efficient and confirm the correctness of the theoretical derivation.