(pdf version)

This two-part talk will consider two geometric integrators, both based on (partitioned) Runge–Kutta.

The first part is a generalisation of the Kahan method, an RK method which preserves an invariant measure and energy for first-order quadratic Hamiltonian systems. The generalisation applies to higher order equations of higher degree, such as cubic second-order Hamiltonian systems. An example, introduced by Reinout Quispel and Andy Hone, is \(x_2-2x_1+x_0 = \epsilon x_0 x_1 x_2\). This approach is extended to ODEs of the form \(x^{(n)} = P_{n+1}(x)\), \(x\in R^m\), finding a first integral for \(n=2\), any \(m\), and an invariant measure for any \(n\) and \(m\). The methods used are a combination of Darboux polynomials and affine equivariance.

The second part concerns a class of methods, originally developed for use in celestial mechanics, formed by doubling the phase space to obtain a separable system, combined with a symmetric projection to the original phase space. I show that this in fact yields a mono-implicit symplectic Runge-Kutta method, prompting the consideration of this class of methods.