New integration methods for perturbed ODEs based on symplectic implicit Runge-Kutta schemes with application to solar system simulations

TL;DR

This paper introduces a new family of symplectic integrators, FCIRK methods, designed for perturbed Hamiltonian ODEs, with applications to efficient long-term solar system simulations.

Contribution

The paper presents FCIRK methods, combining flow composition and implicit Runge-Kutta schemes, offering symplectic, symmetric, and high-order accurate integrators for perturbed Hamiltonian systems.

Findings

Methods are symplectic for Hamiltonian perturbations.

They enable mixed precision implementation for efficiency.

Preliminary experiments show promise in solar system modeling.

Abstract

We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicit Runge-Kutta (IRK) method applied to a transformed system. The resulting integration schemes are symplectic when both the perturbation and the unperturbed part are Hamiltonian and the underlying IRK scheme is symplectic. In addition, they are symmetric in time (resp. have order of accuracy $r$) if the underlying IRK scheme is time-symmetric (resp. of order $r$). The proposed new methods admit mixed precision implementation that allows us to efficiently reduce the effect of round-off errors. We particularly focus on the potential application to long-term solar system simulations, with the equations of motion of the solar system rewritten as a Hamiltonian perturbation of a system of uncoupled Keplerian equations. We present some preliminary numerical experiments with a simple point mass Newtonian 10-body model of the solar system (with the sun, the eight planets, and Pluto) written in canonical heliocentric coordinates.