Search

We give an overview of the construction of algebraic conditions for determining the order of Runge-Kutta methods and describe a novel extension for numerically solving systems of differential equations. The new schemes, called Elementary Differential Runge-Kutta methods, include as a subset Runge-Kutta methods, Taylor series methods, Multiderivative Runge-Kutta methods. We outline how order conditions have been constructed for the new schemes using B-series and their composition and give details relating to a Mathematica implementation. © 2003 Published by Elsevier Science B. V.

This paper focuses on the solution of separable Hamiltonian systems using explicit symplectic integration methods. Strategies for reducing the effect of cumulative rounding errors are outlined and advantages over a standard formulation are demonstrated. Procedures for automatically choosing appropriate methods are also described. © Springer-Verlag 2002.

A component of a new environment for the numerical solution of ordinary differential equations in Mathematica is outlined. We briefly describe how special purpose integration methods can be constructed to solve structured dynamical systems. In particular we focus on the solution of orthogonal matrix differential systems using projection. Examples are given to illustrate the advantages of a projection scheme over conventional integration methods. © Springer-Verlag 2002.