Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators

TL;DR

This paper compares the accuracy and stability of five symplectic and two Runge-Kutta integrators in simulating chaotic Hamiltonian systems, highlighting the limitations of higher-order methods due to Lyapunov instability.

Contribution

It provides a detailed benchmark of multiple integrators on a smooth, chaotic Hamiltonian test case, revealing the practical limitations of higher-order methods.

Findings

Symplectic methods maintain phase volume but are limited by Lyapunov instability.

Higher-order integrators do not necessarily yield more accurate long-term trajectories in chaotic systems.

Double-precision algorithms show varying degrees of accuracy compared to quadruple-precision benchmarks.

Abstract

Time-reversible symplectic methods, which are precisely compatible with Liouville's phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an apparent advantage of such methods. But all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be "accurate". The "advantages" of higher-order methods are lost rapidly for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the two-dimensional one-particle cell model with specially smooth forces. This Hamiltonian problem is chaotic and occurs on a three-dimensional constant-energy shell, the minimum dimension for chaos. We benchmark the problem with quadruple-precision trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and eighth-order Schlier-Seiter-Teloy integrators. We compare the last, most-accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.