Fast Explicit Algorithms for Hamiltonian Numerical Integration

Numerical integration methods for Hamiltonian systems are of importance across many disciplines, including musical acoustics, where many systems of interest are very nearly lossless. Of particular interest are methods possessing a conserved pseu-doenergy. Though most such methods have an implicit character, an explicit method was proposed recently by Marazzato et al. The proposed method relies on a continuous integration which must be performed exactly in order for the conservation property to hold-as a result, it holds only approximately under numerical quadrature. Here, we show an explicit scheme for Hamiltonian integration, with a different choice of pseudoenergy, which is exactly conserved. Most importantly, a fast implementation is possible through the use of structured matrix inversion, and in particular Sherman Morrison inversion of the rank 1 perturbation of a matrix. Applications to the cases of fully nonlinear string vibration, and to the Föppl-von Kármán system describing large amplitude plate vibration are illustrated. Computation times are on par with the simplest non-conservative methods, such as Störmer integration.