High order symplectic integrators for perturbed Hamiltonian systems

TL;DR

This paper introduces a new class of symplectic integrators specifically designed for perturbed Hamiltonian systems, providing higher-order accuracy and improved performance in complex dynamical simulations.

Contribution

The authors develop a constructive method to create symplectic integrators with arbitrary order and positive steps for perturbed Hamiltonian systems, including explicit remainder expressions.

Findings

Integrators achieve order $O( au^p ext{epsilon} + au^2 ext{epsilon}^2)$ with positive steps.

Corrector steps can improve the remainder to $O( au^p ext{epsilon} + au^4 ext{epsilon}^2)$.

Performance tested on simple pendulum and planetary 3-body problem, showing effectiveness.

Abstract

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O(\tau^p\epsilon +\tau^2\epsilon^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O(\tau^p\epsilon +\tau^4\epsilon^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.