On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

TL;DR

This paper demonstrates that second order centered finite differences can be significantly improved to fourth order accuracy on non-uniform grids for boundary value problems with boundary layers, providing insights for more complex applications.

Contribution

It shows how to upgrade the accuracy of finite difference schemes on non-uniform grids for boundary layer problems, highlighting a supraconvergence phenomenon.

Findings

Second order differences can achieve fourth order accuracy on non-uniform grids.

Suitable grid choice is crucial for enhancing finite difference accuracy.

The approach is pedagogical and applicable to more complex boundary value problems.

Abstract

In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.