Local Extrapolation in the Solution of Ordinary Differential Equations

Abstract

The local errors being estimated in the solution of an initial value problem can be added in to make the solution more accurate but this is not always advisable. A rule for deciding when to extrapolate is studied for one-step methods. Some observations about the correctness of local error estimators and extrapolation of multistep methods are also made. Introduction. In a recent paper (1), the author and H. A. Watts studied the effi- ciency of a number of local error estimators for Runge-Kutta methods. Local error is not the only possibility for controlling error in codes for the initial value problem, but it is almost universally used because it is the most practical. There are two other ap- plications of local error estimates which are important. One is to make a locally optimal choice of step size. The other is the subject of this paper; one can add in the estimated error to improve (hopefully) the accuracy of his computations. This local extrapolation asymptotically raises the order of the method by one. Still, extrapola- tion is not necessarily advantageous. The extrapolated value can be less accurate and stability can be seriously affected. We have seen working codes which never extrapolate, always extrapolate, and which provide it as an option. We shall give a simple test which circumvents stability problems for one-step methods and which appears to be effective at choosing to extrapolate only if the accuracy is enhanced. The test is applicable to multistep methods but there are difficulties in this case. We first make some observations about local error estimators and their use in extrapolation. In (1), we were unable to interpret some Runge-Kutta formulas of Zonneveld in terms of local error, but we can interpret them very naturally in this context. We also study two other variants of Runge-Kutta which are candidates for an efficient Runge-Kutta code. Some interesting observations about error estimators for multistep methods are made too. We consider extrapolation of Adams formulas and point out a relation which does not seem as well known as it ought to be. Two rather well-known predictor-corrector pairs are shown by counterexample to have local error estimators which are not asymptotically correct. We provide some com- putations of stability regions for Adams methods which may be of general interest. Local Error Estimators. In (1), we surveyed local error estimators for Runge- Kutta methods and used a set of formulas of Zonneveld (2) to illustrate a different