An analysis of a uniformly accurate spline difference method

Abstract

We are concerned with the numerical approximation by finite difference techniques of the linear two-point boundary value problem and d(x)≥0. Using exponential splines from C ¹[0, 1] as approximation functions and equations with various piecewise constant coefficients as the collocation equations a class of uniformly second order accurate schemes is derived. It is proved for the scheme called IIEMW scheme that the errors at the grid points are bounded by Mh ² whend(x)=0, Mis a constant independent of ∊ and step size h. The numerical results show that the estimate is valid when d(x)≠0. The IIEMW scheme, the well known EI-Mistikawy and Werle (the EMW scheme) and another one, called the IEMW scheme, from the family having second order accuracy, are analysed and compared

Keywords:

• Singular perturbation
• small parameter
• uniform convergency
• spline difference schemes

^∗Corresponding author.

^∗Corresponding author.

Notes

^∗Corresponding author.