Runge-Kutta Methods: Local error control does not imply global error control
This work identifies a fundamental limitation in numerical ODE solvers for practitioners relying on standard error control mechanisms.
The paper demonstrates that local error control via local extrapolation in Runge-Kutta methods does not guarantee global error control, as the global error of the higher-order solution propagates and can cause uncontrolled growth in the lower-order solution's error, often exceeding user-defined tolerances.
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.