Numerical precession in variational discretizations of the Kepler problem
For researchers simulating orbital mechanics, this work offers improved numerical integrators that better preserve the qualitative features of Keplerian orbits.
The paper provides leading-order estimates of numerical precession in the Kepler problem for the implicit MidPoint rule and Störmer-Verlet method, and constructs new integrators that significantly reduce this precession.
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 Störmer-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.