An explicit integrator for the 3D Kepler problem with exact time–true anomaly relation and exact preservation of integrals of motion
Year: 2026
Type: article
Abstract: ABSTRACT We develop and present a numerical scheme for the Kepler problem that preserves exactly all integrals of motion; angular momentum, total energy, and the Laplace–Runge–Lenz vector. This property ensures that orbital trajectories retain their precise shape and orientation over long times, avoiding the spurious precession typical of many standard methods. The scheme uses an adaptive time-step derived from a constant angular increment that allows determining time as a function of the true anomaly up to round-off errors. It means that the induced discretization of the Kepler problem is exact. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability, even for orbits with extreme eccentricity, where other methods fail.
Authors: Jan L Cieśliński, Maciej Jurgielewicz
Institution: University of Białystok
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,046
OpenAlex ID: W7221747524