Stability of step size control based on a posteriori error estimates

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.