Majorant series for the N-body problem

Abstract

As a follow-up of a previous work of the authors, this work considers uniform global time-renormalization functions for the gravitational N-body problem. It improves on the estimates of the radii of convergence obtained therein by using a completely different technique, both for the solution to the original equations and for the solution of the renormalized ones. The aforementioned technique which the new estimates are built upon is known as majorants and allows for an easy application of simple operations on power series. The new radii of convergence so-obtained are approximately doubled with respect to our previous estimates. In addition, we show that majorants may also be constructed to estimate the local error of the implicit midpoint rule (and similarly for Runge-Kutta methods) when applied to the time-renormalized N-body equations and illustrate the interest of our results for numerical simulations of the Solar System.

2010 Mathematics Subject Classifications:

• A34A37
• 70F10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Such an operator is called a Noetherian operator in [17].

Additional information

Funding

M. A. and A. M. have received funding by the Spanish State Research Agency through project PID2019-104927GB-C22 (AEI/FEDER, UE) with acronym GNI-QUAMC, and also from the Department of Education of the Basque Government through the Consolidated Research Group MATHMODE (IT1294-19).