Conservative Integrators for Many-body Problems

TL;DR

This paper develops conservative symmetric second-order integrators for many-body dynamical systems using the Discrete Multiplier Method, ensuring energy preservation and accuracy in complex models like the n-body problem.

Contribution

It introduces a unified approach to construct conservative integrators for various many-body systems, including new schemes for the n-body problem and vortex models.

Findings

Schemes verified to preserve invariants numerically

Achieved second-order accuracy in experiments

Recovered existing conservative schemes for the n-body problem

Abstract

Conservative symmetric second-order one-step schemes are derived for dynamical systems describing various many-body systems using the Discrete Multiplier Method. This includes conservative schemes for the $n$-species Lotka-Volterra system, the $n$-body problem with radially symmetric potential and the $n$-point vortex models in the plane and on the sphere. In particular, we recover Greenspan-Labudde's conservative schemes for the $n$-body problem. Numerical experiments are shown verifying the conservative property of the schemes and second-order accuracy.