Global error estimators for explicit Runge-Kutta methods.

摘要

This study focuses on the construction of explicit Runge-Kutta (RK) methods for estimating the global error by the technique, "Solving for the Correction" considered by Skeel (p.22).;The construction of the RK global error estimators is similar to that of Runge-Kutta methods for the basic problem: parameters of the estimator must satisfy specific conditions to yield a valid error estimate. We develop a strategy for constructing GEEs with particular choices of the arbitrary parameters. Dormand and Prince have obtained some GEEs for RK pairs of orders p and (p - 1) with p = 5 and 6, and each of these can be obtained with the construction developed here. The process also yields some new estimators for RK pairs of higher-orders: p = 7, 8 and 9. For some known RK methods of different orders, standard techniques applied with fixed step sizes to a selected set of problems are used to illustrate that the new error estimators have the orders identified by Dormand and Prince (2).;The theme of this thesis is to examine this construction process both analytically and numerically. (Abstract shortened by UMI.)