Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems

Abstract

The paper is concerned with strongly nonlinear singularly perturbed boundary value problems in one dimension. The problems are solved numerically by finite-difference schemes on special meshes which are dense in the boundary layers. The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed. For the central scheme, error estimates are derived in a discrete $L^1$ norm. They are of second order and decrease together with the perturbation parameter ε. The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically. Numerical results show ε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.