Runge-Kutta-Nyström methods for general second order ODEs with application to multi-body systems
Some Runge-Kutta-Nyström (RKN) type methods for the numerical solution of some particular second order equations are presented. The second order equations under consideration are e.g. those which appear in the simulation of some multi-body systems that have the form \( q'' = M(t,q)^{-1} f(t,q,q') \equiv F(t,q,q')\). Here \(q\) is the state vector and \(M\) is an invertible matrix that usually do not depends on \( q'\). Due to the above form, the computation of \(F(t,q,q')\) for different sets of arguments required in standard RKN methods, simplifies considerably if the variables \((t,q)\) are the same for all sets because in this case the same \( M(t,q)^{-1}\) can be used for all computations. Since the computational cost of a \(q\)-stage is higher than a \(q'\)-stage the author proposes to consider RKN methods with a higher number of \(q'\)-stages than \(q\)-stages for these type of equations. In particular he studies families of 5th order RKN methods with 5 \(q\)-stages and 6 \(q'\)-stages with local error estimation by an embedded pair. The proposed family is construced following the approach of \textit{J. R. Dormand} and \textit{P. J. Prince} [J. Comput. Appl. Math. 6, 19-26 (1980; Zbl 0448.65045)] to derive the 5(4) Runge-Kutta pairs. In order to select a suitable method of the family the available parameters have been choosen so that minimize the local error coefficients of the method. The author also describes the construction of a family of 5th order RKN methods with 5 \(q\)-stages and 9 \(q'\)-stages and provide an optimal method into this family. Finally some numerical experiments are presented to test the efficiency of new methods to achieve a given global error comparing them with the RKN version of the classical DOPR15 of Dormand and Prince. Further some remarks to extend the application of the new methods to other types of equations are given.
scientific article; zbMATH DE number 1311259