Nonlocal convergence analysis of a layer-adaptive ADI scheme for 2D semilinear reaction-subdiffusion equations involving singularity

This article presents nonlocal convergence analysis of a layer adaptive numerical scheme for solving blow-up solutions of two-dimensional semilinear reaction-subdiffusion equations. The fractional derivative is taken in the Caputo sense of order α∈(0,1). In general, the typical solution to such problems exhibit mild singularity near t=0. Firstly, the fractional operator is discretized employing the higher order scheme, namely L2-1σ technique on a layer-adaptive graded mesh in time to investigate the existing mild singularity near the initial time. The classical central difference is used in the uniform partition in space. Using a suitable norm, we prove the stability of the scheme and derive the global error bound. It is established that the proposed scheme achieves a stable and consistent approximation up to twice the fractional order, for a suitable choice of the grading parameter. Then, the proposed scheme is modified and reconstructed to solve the Allen–Cahn type semilinear evolution equations. The Newton’s linearization technique is employed to tackle the nonlinear part. Numerical simulations and comparisons with existing results demonstrate that the derived fully-discrete approach is efficient and effective for solving blow-up solutions of linear and semilinear time-fractional evolution equations. The computational graphs and tabular data indicate that the blow-up behavior of the solutions near the origin intensifies as the order of the fractional derivative increases.