Convergence analysis of a second-order finite difference scheme on Shishkin-type meshes for a singularly perturbed Volterra integro-differential equation

In this paper, a novel second-order numerical method on a Shishkin mesh is constructed to solve a singularly perturbed Volterra integro-differential equation. The proposed numerical scheme employs a second-order backward differentiation formula (BDF2) for discretizing the first-order derivative term, while utilizing the trapezoidal rule to approximate the integral term. Specifically, at the grid transition point, a first-order finite difference approximation is implemented to handle the first-order derivative computation. Subsequently, comprehensive truncation error estimations and rigorous convergence analyses are systematically conducted. Finally, two numerical examples are performed to verify the theoretical findings.