Sharp Error Estimate of Variable Time-Step Imex BDF2 Scheme for Parabolic Integro-Differential Equations with Initial Singularity Arising in Finance

Abstract

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 $\mathcal{O}(N^{− min\{2,γα\}})$ 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 $γ_{{\rm opt}} = 2/α.$ Numerical examples are provided to demonstrate our theoretical analysis.