  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.
