# Symplectic Integrators: T + V Revisited and Round-Off Reduced

## TL;DR

This paper revisits T+V symplectic integrators, enhancing their efficiency and accuracy with higher-order algorithms, variable step sizes, and a novel method to reduce round-off errors, making them competitive with MVS methods for long-term simulations.

## Contribution

The paper introduces new 6th-order T+V algorithms with force gradients and symplectic correctors, and a simple modification to significantly reduce round-off errors in T+V integrators.

## Key findings

- 6th-order T+V algorithms are competitive with MVS in some cases
- Variable step sizes improve T+V efficiency
- Round-off errors can be reduced by several orders of magnitude

## Abstract

Symplectic integrators separate a problem into parts that can be solved in isolation, alternately advancing these sub-problems to approximate the evolution of the complete system. Problems with a single, dominant mass can use mixed-variable symplectic (MVS) integrators that separate the problem into Keplerian motion of satellites about the primary, and satellite-satellite interactions. Here, we examine T+V algorithms where the problem is separated into kinetic T and potential energy V terms. T+V integrators are typically less efficient than MVS algorithms. This difference is reduced by using different step sizes for primary-satellite and satellite-satellite interactions. The T+V method is improved further using 4th and 6th-order algorithms that include force gradients and symplectic correctors. We describe three 6th-order algorithms, containing 2 or 3 force evaluations per step, that are competitive with MVS in some cases. Round-off errors for T+V integrators can be reduced by several orders of magnitude, at almost no computational cost, using a simple modification that keeps track of accumulated changes in the coordinates and momenta. This makes T+V algorithms desirable for long-term, high-accuracy calculations.