Elementary Tutorial on Numerical Methods for Singular Perturbation Problems

Abstract

In the first section we introduce a simple singularly perturbed initial value problem for a first order linear differential equation. We construct the backward Euler finite difference method for this problem. We then discuss continuous and discrete maximum principles for the associated continuous and discrete operators and we conclude the section by defining what is meant by a parameter-uniform numerical method. In the second section we introduce a fitted operator method on a uniform mesh for our simple initial value problem defined in the previous section. We then prove rigorously that this method is parameter-uniform at the mesh points. Fitted mesh methods on piecewise uniform meshes are introduced in the third section. A fitted mesh method for our simple initial value problem is constructed. It is proved rigorously that this method is parameter-uniform at the mesh points. Finally, in the fourth section, numerical solutions of singular perturbation problems are discussed. Computations using standard and a parameter-uniform numerical method are presented. The usefulness and reliability of parameter-uniform methods is demonstrated.