Two new optimal and uniform third-order schemes for singular perturbation problems with initial layers

Abstract

This article presents two numerical methods of the order of three for singular perturbation problems, with a small positive parameter using finite differences. It is a problem with an initial layer in the neighborhood of the initial nodal point whose width is of the order of the small parameter $\epsilon$. Explicit order three classical methods are modified, and a new scheme is designed for singular perturbation problems. It is a fitted operator method and it is explicit with a variable fitting factor (VFF) evaluated at all nodal points. To reduce the calculation time of the scheme with VFF, the VFF is replaced by a fixed constant fitting factor (CFF). It is implicit with a CFF which is evaluated only one time at the initial nodal point. These two schemes are both optimal concerning the small parameter $\epsilon$ and uniform of order three. The three order methods presented in this article are superior to the three order methods available in the literature. To view the initial layer when the mesh size is larger than the parameter in the problem, these two fitted operator methods are extended to fitted mesh methods since fitted mesh methods are layer rescaling. The construction of the fitted mesh method is also provided. That is, the uniform mesh is extended to non-uniform mesh. Experimental results are presented to show the optimal and higher-order performance of the two numerical methods using three test problems.