Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

TL;DR

This paper develops and analyzes geometric integration schemes for dissipative Hamiltonian systems with Rayleigh damping, demonstrating their energy dissipation accuracy and near conservation properties through theoretical analysis and numerical examples.

Contribution

It introduces geometric integrators that preserve structure for perturbed Hamiltonian systems with damping and proves their asymptotic energy dissipation and stability properties.

Findings

Energy dissipation rate is asymptotically correct.

Modified Hamiltonian decreases monotonically for small step sizes.

Numerical examples show superior performance over Runge-Kutta methods.

Abstract

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the symplectic structure in the case $\epsilon=0$. In the case $0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order.