Numerical precession in variational discretizations of the Kepler problem

TL;DR

This paper analyzes the numerical precession observed in simulations of the Kepler problem and introduces new integrators that reduce this precession by leveraging Lagrangian structure and perturbative Noether's theorem.

Contribution

It provides leading order estimates of precession for MP and SV methods and develops new integrators with improved accuracy for the Kepler problem.

Findings

Leading order precession estimates for MP and SV methods

New integrators outperform MP and SV in reducing precession

Enhanced numerical stability and accuracy in Kepler simulations

Abstract

Kepler's first law states that the orbit of a point mass with negative energy in a classical gravitational potential is an ellipse with one of its foci at the gravitational center. In numerical simulations of this system one often observes a slight precession of the ellipse around the gravitational center. Using the Lagrangian structure of modified equations and a perturbative version of Noether's theorem, we provide leading order estimates of this precession for the implicit MidPoint rule (MP) and the St\"ormer-Verlet method (SV). Based on those estimates we construct some new numerical integrators that perform significantly better than MP and SV on the Kepler problem.