A Posteriori Error Estimator for a Non-Standard Finite Difference Scheme Applied to BVPs on Infinite Intervals

TL;DR

This paper develops an a posteriori error estimator for a non-standard finite difference scheme applied to boundary value problems on infinite intervals, utilizing Richardson's extrapolation to improve accuracy and estimate errors.

Contribution

It introduces a novel a posteriori error estimator using Richardson's extrapolation for non-standard finite difference schemes on infinite domains.

Findings

Richardson's extrapolation provides an effective upper bound for the global error.

The estimator is validated on a benchmark problem with a known exact solution.

The method is effective when round-off errors are negligible and grids are sufficiently fine.

Abstract

In this paper, we present a study of an a posteriori estimator for the discretization error of a non-standard finite difference scheme applied to boundary value problems defined on an infinite interval. In particular, we show how Richardson's extrapolation can be used to improve the numerical solution involving the order of accuracy and numerical solutions from two nested quasi-uniform grids. A benchmark problem is examined for which the exact solution is known and we get the following result: if the round-off error is negligible and the grids are sufficiently fine then the Richardson's error estimate gives an upper bound of the global error.