The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms

TL;DR

This paper develops a method to analytically compute perihelion precession errors caused by symplectic integrators in the Kepler problem, revealing how symplectic correctors can cancel these errors for improved long-term accuracy.

Contribution

It introduces a general approach to calculate perihelion advance errors for symplectic integrators, including non-separable Hamiltonians, and explains how corrector algorithms eliminate these errors.

Findings

Error Hamiltonians cause opposite precession at each order.

Symplectic correctors cancel precession errors after each period.

Optimal algorithms are determined by the physics of symplectic integrators.

Abstract

Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for non-separable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at each order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors exactly cancel after each period. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions.