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The recently developed technique of DOC kernels has been a great success inthe stability and convergence analysis for BDF2 scheme with variable timesteps. However, such an analysis technique seems not directly applicable toproblems with initial singularity. In the numerical simulations of solutionswith initial singularity, variable time-steps schemes like the graded mesh arealways adopted to achieve the optimal convergence, whose first adjacenttime-step ratio may become pretty large so that the acquired restriction is notsatisfied. In this paper, we revisit the variable time-step implicit-explicittwo-step backward differentiation formula (IMEX BDF2) scheme presented in [W.Wang, Y. Chen and H. Fang, \emph{SIAM J. Numer. Anal.}, 57 (2019), pp.1289-1317] to compute the partial integro-differential equations (PIDEs) withinitial singularity. We obtain the sharp error estimate under a mildrestriction condition of adjacent time-step ratios $r_{k}: =\tau_{k}/\tau_{k-1}\; (k\geq 3) < r_{\max} = 4.8645 $ and a much mild requirement on the firstratio, i.e., $r_2>0$. This leads to the validation of our analysis of thevariable time-step IMEX BDF2 scheme when the initial singularity is dealt by asimple strategy, i.e., the graded mesh $t_k=T(k/N)^{\gamma}$. In thissituation, the convergence of order $\mathcal{O}(N^{-\min\{2,\gamma \alpha\}})$is achieved with $N$ and $\alpha$ respectively representing the total meshpoints and indicating the regularity of the exact solution. This is, theoptical convergence will be achieved by taking $\gamma_{\text{opt}}=2/\alpha$.Numerical examples are provided to demonstrate our theoretical analysis.

Sharp error estimate of variable time-step IMEX BDF2 scheme for parabolic integro-differential equations with initial singularity arising in finance