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

Abstract

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