Benchmark: Tao's symplectic integration method

TL;DR

This paper evaluates Tao's symplectic integration method through benchmark tests, comparing its performance to established methods like Störmer-Verlet and Runge-Kutta, highlighting its advantages in generality and ease of implementation.

Contribution

The paper provides a comprehensive benchmark analysis of Tao's explicit symplectic integrator, demonstrating its comparable performance and greater versatility over traditional methods.

Findings

Tao's method is similar in speed to Störmer-Verlet.

It is slower than Runge-Kutta-Cash-Karp but more general.

The method conserves system invariants effectively.

Abstract

A benchmark test was conducted for a new symplectic integration method originally developed by Molei Tao. The method raises interest due to its explicit evolution equation, with applicability to both separable and non-separable Hamiltonian systems, and an easy-to-implement, easily generalizable algorithm. In order to compare the method with other, more well-known methods, namely St\"{o}rmer-Verlet and Runge-Kutta, we conducted a series of benchmark tests comparing their performance in terms of CPU time, system invariants functions conservation, and numerical symplectic area conservation. Overall, it was found that despite being slower than the more optimized Runge-Kutta-Cash-Karp, Tao's method presents a similar performance to St\"{o}rmer-Verlet, with the extra perk of being more generic and not requiring the use of implicit equations for the evolution of the equations of motion.