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.

翻译：基于残差的后验误差估计可用于数值方法的可靠误差控制。本文在常微分方程和龙格-库塔方法的背景下考虑此类估计，特别采用Dedner & Giesselmann (2016)的方法并研究其用于选择时间步长时的表现。我们重点分析结合显式龙格-库塔方法时的步长控制稳定性，证明标准I控制器不稳定，而更先进的PI和PID控制器可被设计为稳定的。通过理论分析与数值实验，我们比较了基于残差的估计器与基于嵌入龙格-库塔方法的经典误差估计器的稳定性性质。