Higher order splitting methods with modified integrators for a class of Hamiltonian systems

TL;DR

This paper develops higher order splitting methods with positive intermediate steps for Hamiltonian systems, improving accuracy beyond the standard second order method.

Contribution

It introduces systematic higher order splitting schemes with real, positive intermediate steps for a broad class of Hamiltonian systems, extending the standard Verlet method.

Findings

Achieves accuracy of order τ^4, τ^6, and τ^8

All intermediate timesteps are real and positive

Applicable to a large class of Hamiltonian systems

Abstract

We discuss systematic extensions of the standard (St{\"o}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics, with relative accuracy of order $\tau^2$ for a timestep of length $\tau$, to higher orders in $\tau$. We present some splitting schemes, with all intermediate timesteps real and positive, which increase the relative accuracy to order $\tau^{N}$ (for N=4, 6, and 8) for a large class of Hamiltonian systems.