Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy

Abstract

A numerical study is made for a class of boundary value problems of second-order differential equations in which the highest order derivative is multiplied by a small parameter ϵ and both the differentiated(convection) and undifferentiated(reaction) terms are with negative shift δ. We analyze three difference operators L k h , k = 1, 2, 3 a simple upwind scheme, midpoint upwind scheme and a hybrid scheme, respectively, on a Shishkin mesh to approximate the solution of the problem. Theoretical error bounds are established. The accuracy by one of the grid adaptation strategies, namely grid redistribution is examined by solving the considered problem using hybrid method for some values of ϵ and N. As a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost. A few numerical results exhibiting the performance of these three schemes are presented.