# Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

## TL;DR

This paper introduces a near-optimal implementation of symplectic implicit Runge-Kutta schemes for Hamiltonian systems, focusing on reducing and monitoring round-off error propagation during high-precision numerical integration.

## Contribution

The paper presents a novel implementation that minimizes round-off error propagation in symplectic implicit Runge-Kutta schemes using fixed point iteration, with an error estimation procedure.

## Key findings

- Implementation is near-optimal in round-off error propagation.
- Error estimation method effectively monitors propagation.
- Numerical experiments validate the approach.

## Abstract

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.