Chengchao Zhao, Ruoyu Yang, Yana Di, Jiwei Zhang
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 $r_k:=\tau_k / \tau_{k-1}<r_{\text {max }}=4.8645(k \geq 3)$ 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 order of $\mathcal{O}\left(N^{-\min \{2, \gamma \alpha\}}\right)$ is achieved, where $N$ denotes the total number of mesh points and $\alpha$ indicates the regularity of the exact solution. This is, the optimal convergence will be achieved by taking $\gamma_{\text {opt }}=2 / \alpha$. Numerical examples are provided to demonstrate our theoretical analysis.
