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

The recently developed technique of DOC kernels has been a great success in the stability and convergence analysis for BDF2 scheme with variable time steps. However, such an analysis technique seems not directly applicable to problems with initial singularity. In the numerical simulations of solutions with initial singularity, variable time-steps 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 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) with initial singularity. We obtain the sharp error estimate under a mild restriction 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 first ratio, i.e., $r_2>0$. 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 $t_k=T(k/N)^{\gamma}$. In this situation, the convergence of order $\mathcal{O}(N^{-\min\{2,\gamma \alpha\}})$ is achieved with $N$ and $\alpha$ respectively representing the total mesh points and indicating the regularity of the exact solution. This is, the optical convergence will be achieved by taking $\gamma_{\text{opt}}=2/\alpha$. Numerical examples are provided to demonstrate our theoretical analysis.