Stage Value Predictors and Efficient Newton Iterations in Implicit Runge--Kutta Methods

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Abstract
The prediction of stage values in implicit Runge--Kutta methods is important both for overall efficiency as well as for the design of suitable control strategies for the method. The purpose of this paper is to construct good stage value predictors for implicit methods and to verify their behavior in practical computations. We show that for stiffly accurate methods of low stage order it is necessary to use several predictors. In other words, a continuous extension for the method will not yield the best results. We also investigate how to gain additional efficiency in the Newton iterations used to correct the prediction error. This leads to new control strategies with respect to refactorization of Jacobians that seek to globally minimize total work per unit time of integration.