Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods

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Abstract
The problem of stepsize selection in the numerical solution of ordinary differential equations can be viewed as an automatic control problem. We will demonstrate how control theory can be used to analyze and improve the standard stepsize control algorithm. Previously, Gustafsson et al. (5) suggested a PI controller to overcome the problem of oscillating stepsize sequences that typically appear when explicit Runge-Kutta methods en- counter stiffness. Its properties were investigated experimentally. Here, the superior properties of the PI controller will be analyzed using a model for the relation between stepsize and the local truncation error in the integration method. When stability limits the stepsize, the standard asymptotic model fails to correctly describe this relation. Instead, a dynamic model that takes this behavior into account is derived for explicit Runge-Kutta methods. The model is verified using numerical tests and system identification. The derived model helps analyzing standard stepsize control. The analysis gives insight and leads to a good understanding of the properties of the control system. The acquired understand- ing is used to further improve the PI controller as well as tuning its parameters,

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