Numerical time-stepping methods which preserve geometric properties of time-symmetric and Hamiltonian problems over very long times. In particular, the analysis and construction of general linear methods (GLMs), which are a generalization of Runge-Kutta and linear multistep methods.

Recent work with Prof John Butcher (University of Auckland) and Terence Norton (PhD student, Bath) has shown that G-symplectic time-symmetric GLMs can solve non-separable Hamiltonian systems more efficiently than symplectic Runge- Kutta methods. Methods up to order 6 have been constructed. The underlying one-step method of such methods has been shown to be both symmetric and symplectic.

A study of starting and finishing methods has led to a re-evaluation of the way that classical numerical methods, such as Leapfrog and Crank-Nicolson, should be implemented on symplectic and near symplectic systems, such as the equations modelling weather.

Other interests include nonautonomous stability for both continuous and discrete dynamical systems.