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

Abstract

The problem of stepsize selection in implicit Runge-Kutta schemes is analyzed from a feedback control point of view. This approach leads to a better understanding of the relation between stepsize and error. A new dynamical model describing this relation is derived. The model is used as a basis for a new stepsize selection rule. This rule achieves better error control at little extra expense. The properties of the new model and the improved performance of the new error control are demonstrated using both analysis and numerical examples.