Error Estimation of Numerical Solvers for Linear Ordinary Differential Equations

TL;DR

This paper introduces a new approach for estimating the global error in numerical solutions of linear ODEs using residuals based on Hermit interpolation, enhancing the reliability assessment of these methods.

Contribution

It provides a novel definition of residuals for ODE solvers and establishes a bound for the global error, improving error estimation techniques.

Findings

Residual-based error estimation is effective for various ODE models.

The proposed bounds accurately reflect the true global error.

Method enhances the reliability assessment of numerical ODE solvers.

Abstract

Solving Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions. However, few work about global error estimation can be found in the literature. In this paper, we first give a definition of the residual, based on the piecewise Hermit interpolation, which is a kind of the backward-error of ODE solvers. It indicates the reliability and quality of numerical solution. Secondly, the global error between the exact solution and an approximate solution is the forward error and a bound of it can be given by using the backward-error. The examples in the paper show that our estimate works well for a large class of ODE models.