Difference approximations for singular perturbations of systems of ordinary differential equations

Summary

We consider difference methods for the solution of singular perturbations of boundary value problems. The solutions are smooth except in boundary layers of thickness ε|logε|, 0<ε≪1. Various difference schemes with a uniform stepsizeh are considered. In practice,h is usually much larger than the boundary layer regions. Then the difference methods must be choosen with care. It is shown that only approximations of low order accuracy can be used. However, one can increase the accuracy by a Richardson procedure. Asymptotic expansions in powers ofh and ε are given for the solutions of the proposed methods.