Special Finite Difference Method for Singular Perturbation Problems with One-End Boundary Layer

K Phaneendra,Y.N Reddy,K Madhulatha

doi:10.3390/mca19030208

Abstract

In this paper, we have presented a special finite difference method for solving a singular perturbation problem with layer behaviour at one end. In this method, we have used a second order finite difference approximation for the second derivative, a modified second order upwind finite difference approximation for the first derivative and a second order average difference approximation for y to reduce the global error and retaining tridiagonal system. Then the discrete invariant imbedding algorithm is used to solve the tridiagonal system. This method controls the rapid changes that occur in the boundary layer region and it gives good results in both cases i.e., h ≤ ε and ε << h. The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method. We have presented maximum absolute errors for the standard examples chosen from the literature.

Highlights

Perturbed boundary value problems arise frequently in many areas of science and engineering such as heat transfer problem with large Peclet numbers, Navier–Stokes flows with large Reynolds numbers, chemical reactor theory, aerodynamics, reaction–diffusion process etc. due to the variation in the width of the layer with respect to the small perturbation parameter ε
Several difficulties are experienced in solving the singular perturbation problems using standard numerical methods
We have described a special finite difference method for solving a singular perturbation problem with layer behaviour at on end point

Summary

INTRODUCTION

Perturbed boundary value problems arise frequently in many areas of science and engineering such as heat transfer problem with large Peclet numbers, Navier–Stokes flows with large Reynolds numbers, chemical reactor theory, aerodynamics, reaction–diffusion process etc. due to the variation in the width of the layer with respect to the small perturbation parameter ε. The numerical treatment of singularly perturbed differential equations gives major computational difficulties due to the presence of boundary and/or interior layers This type of problem was solved asymptotically by Bellman [1], Bender and Orszag [2], Kevorkian and Cole [3], Nayfeh [4], O’Malley [5] and numerically by Kreiss [6], Miller [7], Kadalbajoo and Devendra Kumar [8], Reddy [9, 10], Lin and Vancouver [11], and Van Veldhuizen [12] etc. Reddy removed by approximating the first derivative by second order modified upwind difference

Left-end boundary layer problems

STABILITY AND CONVERGENCE ANALYSIS

NUMERICAL EXAMPLES

DISCUSSIONS AND CONCLUSIONS