Stability of step size control based on a posteriori error estimates

TL;DR

This paper investigates the stability of step size control in explicit Runge-Kutta methods using residual-based a posteriori error estimates, showing that advanced controllers can ensure stability while standard ones may not.

Contribution

It demonstrates the stability properties of residual-based error estimators and designs stable PI and PID controllers for step size selection in explicit Runge-Kutta methods.

Findings

Standard I controllers are unstable with residual-based estimates.

PI and PID controllers can be designed for stability.

Residual-based estimators have favorable stability properties.

Abstract

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.