This work focuses on the derivation of composition methods for the numerical integration of ordinary differential equations, which give rise to very challenging optimization problems. Composition is a useful technique for constructing high order approximations, while conserving certain geometric properties. We survey existing composition methods and describe results of an intensive numerical search for new methods. Details of the search procedure are given along with numerical examples, which indicate that the new methods perform better than previously known methods. Some insight into the location of global minima for these problems is obtained as a result.

Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. Runge-Kutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods, involving only s(s + 1)/2 free parameters for a symplectic sstage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrized coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In this case the order conditions associated with even trees become redundant.

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Order stars are a powerful modern tool for the development and analysis of numerical methods. They convey important information such as order and stability in a unified framework. A package for rendering order stars becomes part of the standard distribution in the next major release of Mathematica. An introduction to the theory is provided here, set in the context of numerical methods for Ordinary Differential Equations. The implementation is discussed and examples are given to illustrate why a computer algebra system is an ideal environment for the exploration of order stars.

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