Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping
For researchers in computational dynamics, this provides theoretically grounded numerical methods for dissipative perturbations of Hamiltonian systems.
The paper analyzes explicit and semi-explicit geometric integration schemes for Hamiltonian systems with small Rayleigh damping, proving asymptotic correctness of energy dissipation and near conservation of relative equilibria. Numerical examples show superior performance over conventional Runge-Kutta methods.
Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $ε$, and the schemes under study preserve the symplectic structure in the case $ε=0$. In the case $0<ε\ll 1$ 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.