Abstract
Integrating Hamiltonian dynamical systems over long integration times can be difficult due
to the need to preserve the underlying symplectic structure and geometric properties such as
energy and momentum. Traditional numerical integrators often introduce artificial damping,
leading to the gradual drift of these conserved properties, which are fundamental to the accuracy
of the numerical simulation. Symplectic integrators, a particular class of numerical integrators
specifically developed to preserve these structural properties of ordinary differential equations
(ODEs) that model Hamiltonian dynamical systems, have become an important tool to tackle
this issue.
This study demonstrates that symplectic integrators often outperform traditional non-symplectic
numerical integrators when solving Hamiltonian ODEs. The study establishes through numerical
experiments on well-known Hamiltonian dynamical systems that symplectic integrators
preserve “almost exactly” the system’s energy, with the associated energy error bounded above
by O(hp) over an exponentially long integration time. In contrast, the study shows that nonsymplectic
numerical integrators demonstrate an unbounded growth in energy error. Additionally,
the results of this study show that even lower-order symplectic integrators can provide
better long-term accuracy than higher-order non-symplectic integrators when applied to Hamiltonian
ODE problems. These findings validate symplectic methods as robust and reliable numerical
integrators for simulating dynamical systems over long periods.