A Uniformly Convergent Difference Scheme for the Singularly Perturbed Periodic Problem

In this article, a novel numerical scheme is suggested to solve periodical boundary value problem for linear first order singularly perturbed equation. This scheme is constructed by the finite difference method on a special non-uniform mesh (Shishkin mesh) using quadrature rules with the remaining terms in integral form. It is proven that the scheme achieves almost first-order convergence on the discrete maximum norm. Finally, two test problems are considered to demonstrate the accuracy and performance of the method.