Cheap Error Estimation for Runge--Kutta Methods

• Authors: CH. Tsitouras;S. N. Papakostas
• Venue: SIAM Journal on Scientific Computing
• Year: 1999

Explicit Runge--Kutta (RK) methods in the form of pairs of orders p(p - 1) provide an attractive means for the solution of initial value problems of first-order differential equations. Most existing RK formulas (single methods as well as pairs) use the minimal number of stages required for achieving a prescribed order. In this article we shall study, in terms of efficiency and reliability, RK pairs of orders p( q), whenever q p-1. While in practice pairs of orders p( p - 1) usually require one or two more stages in addition to those already necessary for a pth-order single method, we show here that if p = 6, 7, 8, or 10, then efficient pairs of orders p(p - 2), p(p - 3), or p(p - 4) may be easily constructed with a reduced cost in function evaluations with respect to pairs of orders p(p - 1). In general comparing p(qt) pairs used in local extrapolation mode (LEM---the one most frequent in practice), we see that while the propagated solution of a problem is, in either case, of the same order p, pairs characterized by q p - 1 use fewer function evaluations. Consequently they might be more efficient, provided that they are accompanied by a reliable estimator, and an efficient implementation could be found for their application in practical situations. A new step-size selection algorithm proposed here takes full advantage of the potential for increased efficiency inherited by pairs that are accompanied by no-additional-cost estimators. This algorithm, which may be also applied to Nystrom pairs, makes code implementation of these pairs attractive, as in all cases the proportionality of the global error with respect to the requested tolerance is in practice always achieved. Here we shall cover the cases of pairs of orders 4(2), 6(4), 7(5), 8(6), 8(5), 8(4), and 10(6) with nearly minimized truncation error coefficients which use a minimal number of stages (i.e., in almost all cases equal to that currently known to be the minimal required for constructing a single method of order equal to that of the higher-order method of the pair). By studying the numerical performance of these pairs we may see that not only are these pairs as reliable as the respective pairs of the type p(p - 1), but in all cases they seem to be more efficient. Another important consequence of the numerical tests performed here is that they suggest that for a given number of stages the best RK pairs that may be attained are those for which the higher-order method is of the maximal possible order.