Variable stepsize schemes for effective order methods and enhanced order composition methods (Q5934372)

Two approaches to circumvent the Butcher barriers on the maximum order of explicit Runge-Kutta methods are examined and some numerical examples presented. The first approach is based on the concept of effective order introduced by Butcher in 1969 [for a more recent reference see \textit{J. C. Butcher}, The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, Wiley, New York (1987; Zbl 0616.65072)]. This author proved that there exist e.g. explicit Runge-Kutta methods \( \alpha \) with six stages such that for a suitable method \(\beta\) the composition \( \beta \alpha \beta^{-1}\) has order six (this is the so called effective order of \(\alpha\)) while the maximum order of any explicit Runge-Kutta method with six stages is five. Moreover a repeated application of \( \beta \alpha \beta^{-1}\) with a fixed step size: \( (\beta \alpha \beta^{-1})^n \equiv \beta \alpha^n \beta^{-1}\) can be computed, apart of the starting and ending adjustment, with the same computational cost as the \( \alpha \) method. In the paper under consideration the authors extend the idea of effective order to the case of variable step size and illustrate their approach by constructing a variable step size Runge-Kutta method with five stages and effective order five which includes also local error estimation. The main difficulty of the variable step size effective order, from a practical point of view, is that the coefficients of the method depend on the step size ratio and therefore must be recalculated at every step size change. The second approach consist in combining successively two different Runge-Kutta methods of the same order \( p\) so that they possess the same leading error term with opposite sign and therefore they behave globally as a method of order \( \geq p+1 \). With this idea the authors construct two explicit methods with six stages of order five whose composition has order six. Finally some numerical experiments are presented to show that the new methods can be competitive with other well known methods in use.