Numerical Solution of Singularly Perturbed Boundary Value Problems with Twin Boundary Layers using Exponential Fitted Scheme

Abstract

This paper deals with a numerical method with fitted operator difference method for twin (dual) boundary layers singularly perturbed boundary value problems. In this method, Numerov method is extended to the given second order problem having derivative of first order. Using the non standard differences and modified upwind difference for the first order derivatives, the discrete scheme is deduced. A fitting parameter is utilized in the difference scheme, which handles the rapid changes that occur in the boundary layers due to the small perturbation parameter. Tridiagonal solver is implemented to solve the system of the method. Convergence analysis of the deduced method is discussed. Maximum errors in the solution of the model numerical examples are tabulated and comparison is made, to illustrate and support the method. Solutions are depicted graphically to show the layer behaviour.

Absolutely stable difference scheme for a general class of singular perturbation problems

“…The results in Tables 1, 2 and 3 show that at δ i = 4/ω|p i | or δ i = δ op = 4/ω|p i |, there is no propagation error with the Thomas algorithm (20). Also, at ε = 10 -3 the ASCD is reduced to the CFD scheme, where δ op = 2h, and the results are identical.…”

“…Also, on the other hand, for the right-end boundary layers, p i > 0, we have G i = 0, and the present scheme is reduced to a stable backward integration scheme. Moreover, in the above two cases, the propagation error in (20) has vanished.…”