(untitled) (Journal of Computational Mathematics, 2025)
The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes. However, it may not be readily applicable to problems exhibiting an initial singularity. In the numerical simulations of solutions with initial singularity, variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme to solve the parabolic integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_k := τ_k/τ_k−1 0. This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e. the graded mesh t_k= T (k/N)^γ. In this situation, the convergence order of O(N^− min{2,γα ) is achieved, where N denotes the total number of mesh points and α indicates the regularity of the exact solution. This is, the optimal convergence will be achieved by taking γ_opt = 2/α. Numerical examples are provided to demonstrate our theoretical analysis.