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.

In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.