A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

A novel parameter-uniform finite difference scheme on a Shishkin-type mesh for a singularly perturbed Volterra integro-differential equation is studied. The problem is discretized by the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is carried out. It is shown from the convergence analysis that our presented method is almost second-order uniformly convergent with respect to the perturbation parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} in the discrete maximum norm. Numerical results are given to support our theoretical result.