Revisiting high-order Taylor methods for astrodynamics and celestial mechanics

We present heyoka, a new, modern and general-purpose implementation of Taylor's integration method for the numerical solution of ordinary differential equations. Detailed numerical tests focused on difficult high-precision gravitational problems in astrodynamics and celestial mechanics show how our general-purpose integrator is competitive with and often superior to state-of-the-art specialised symplectic and non-symplectic integrators in both speed and accuracy. In particular, we show how Taylor methods are capable of satisfying Brouwer's law for the conservation of energy in long-term integrations of planetary systems over billions of dynamical timescales. We also show how close encounters are modelled accurately during simulations of the formation of the Kirkwood gaps and of Apophis' 2029 close encounter with the Earth (where heyoka surpasses the speed and accuracy of domain-specific methods). heyoka can be used from both C++ and Python, and it is publicly available as an open-source project.