Symbolic derivation of Runge-Kutta methods
This paper is concerned with a symbolic Mathematica package Butcher.m which automates the derivation of order conditions of Runge-Kutta (RK) methods for the numerical solution of ordinary differential equations. It must be noticed that there is a first version of Butcher.m written by Keiper around 1988. However the version under consideration here is new and incorporates substantial improvements not only in the derivation of the order conditions but also in the possibility to use different sets of simplifying assumptions to get more easily different families of RK formulas. The paper explains the techniques used by the author to generate efficiently the set of all rooted trees and the elementary weights and densities associated to the trees which are essential to derive the order conditions in the RK methods. It follows from the remarks of the author that computer algebra has been considerably reduced over the first version of Butcher.m. Next, the problem of solving the set of order conditions for a given number of stages and order is considered. At this point the use of simplifying assumptions and the more general technique of Gröbner bases is examined. Some examples are presented to show the potentiality of this package. In conclusion we have a new tool which can be very useful in the investigation of RK schemes, thus for a given formula (even of higher order) we may compute straightforwardly the coefficients of the elementary differentials in the local error. However, due to the complexity, the derivation of e.g. high order RK formulas (or pairs of) is far from being solved with this package.
scientific article; zbMATH DE number 740428