A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation

We consider a linear singularly perturbed Volterra integro-differential equation. Our aim is to design and analyse a finite difference method which is robust with respect to the perturbation parameter to solve this equation. The method we construct is a combination of backward Euler difference operator for the differential part and repeated quadrature rules for the integral part. We show that the method is the first-order convergent in the maximum norm. Numerical experiments are carried out on some test examples, confirming the robustness of the proposed scheme.