Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm

Abstract

In this article, we study the numerical solution of singularly perturbed non-linear autonomous initial-value problems by a non-standard algorithm on layer-resolving nonuniform meshes. Here, we use the piecewise-uniform Shishkin meshes, and two other nonuniform meshes which resolve the difficulties arising from the steep gradient of the solution in the initial layer. The present method is intended for solving the nonlinear problem without linearization and provides third-order convergence results. Linear stability of this method is studied. Numerical experiments are carried out to verify the efficiency and accuracy of the method.