Dissertation

Symplectic Integrators

Sibelo, Godknows

2024

Overview

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.

• Publisher: ProQuest Dissertations & Theses
• Subject: Dynamical systems / Energy conservation / Geometry / Lie groups / Mathematics / Mechanics / Ordinary differential equations / Textbooks
• ISBN: 9798273357693