High order parameter uniform numerical method for singular perturbation problems

In this paper, we systematically describe how to derive a method of higher order for the numerical solution of singularly perturbed ordinary differential equations. First we apply this idea to derive a fourth-order method for a self-adjoint singularly perturbed two point boundary value problem. This method is uniformly convergent on a piecewise uniform mesh of Shishkin type. After we have developed and analyzed a fourth-order method, we explain with appropriate details, how can one obtain the methods of order higher than four which looks straightforward but has not been seen in the literature so far. Besides these, the fourth-order ε-uniformity in the theoretical estimate has been justified by some numerical experiments.