High Order Three Part Split Symplectic Integration Schemes

TL;DR

This paper develops and compares high order symplectic integration schemes for Hamiltonian systems split into three parts, demonstrating their superior long-term energy conservation and efficiency over non-symplectic methods.

Contribution

It introduces novel high order symplectic schemes for three-part split Hamiltonian systems and evaluates their performance against existing methods.

Findings

New symplectic schemes outperform non-symplectic methods in energy conservation.

The schemes are more computationally efficient for long-term simulations.

Effective for both simple models and complex nonlinear equations.

Abstract

Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for a few special cases. In this work, we present and compare different ways to construct high order symplectic schemes for general Hamiltonian systems that can be split in three integrable parts. We use these techniques to numerically solve the equations of motion for a simple toy model, as well as the disordered discrete nonlinear Schr\"odinger equation. We thereby compare the efficiency of symplectic and non-symplectic integration methods. Our results show that the new symplectic schemes are superior to the other tested methods, with respect to both long term energy conservation and computational time requirements.