Abstract:

A singularly perturbed Volterra integro-differential equation is considered. The problem is discretized by using a simple first-order finite difference scheme on a Vulanović-Bakhvalov mesh, the accuracy of which is first-order uniformly convergent with respect to the perturbation parameter $\varepsilon$. Furthermore, based on the Richardson extrapolation technique, the $\varepsilon$-uniform accuracy of the presented approximation scheme can be improved from $O(N^{-1})$ to $O(N^{-2})$, where $N$ is the number of mesh intervals. Finally, the theoretical finds are illustrated by two numerical experiments.

Key words: singularly perturbed, Volterra integro-differential equation, Richardson extrapolation, Vulanovi?-Bakhvalov mesh