Runge-Kutta Methods: Local error control does not imply global error control

TL;DR

This paper demonstrates that controlling local error in Runge-Kutta methods does not guarantee control over the global error, highlighting the need for separate strategies for global error management in numerical ODE solutions.

Contribution

The paper provides a theoretical analysis showing the disconnect between local and global error control in Runge-Kutta methods and identifies conditions affecting global error propagation.

Findings

Local error control does not ensure global error control.

Global error can grow despite local error management.

Conditions for initial global error control are identified.

Abstract

We study the relationship between local and global error in Runge-Kutta methods for initial-value problems in ordinary differential equations. We show that local error control by means of local extrapolation does not equate to global error control. Our analysis shows that the global error of the higher-order solution is propagated under iteration, and this can cause an uncontrolled increase in the global error of the lower-order solution. We find conditions under which global error control occurs during the initial stages of the RK integration, but even in such a case the global error is likely to eventually exceed the user-defined tolerance.