Mathematical Modeling with Multidisciplinary Applications

CHAPTER 14

EFFICIENT NUMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS

S. NATESAN

Department of Mathematics, Indian Institute of Technology Guwahati, India

14.1 INTRODUCTION

Singular perturbation problems (SPPs) arise in several branches of applied mathematics which include fluid dynamics, quantum mechanics, elasticity, chemical reactor theory, gas porous electrodes theory, etc. The presence of small parameter(s) in these problems prevents us from obtaining satisfactory numerical solutions. It is a well known fact that the solutions of SPPs have a multiscale character. That is, there are thin layer(s) where the solution varies very rapidly, while away from the layer(s) the solution behaves regularly and varies slowly. Even in the case where only the approximate solution of the singularly perturbed boundary-value problem is required, classical numerical methods, such as finite difference schemes and finite element methods exhibit unsatisfactory behavior. This arises because the accuracy of the approximate solution depends inversely on the perturbation parameter value and thus it deteriorates as the parameter decreases. Therefore, the numerical treatment of SPPs gives major computational difficulties.

Various finite difference schemes have been proposed in the literature to guarantee stability of the schemes for all values of the perturbation parameter. Careful examination of numerical results from such schemes on uniform grids shows that, for fixed (small) values ...