New insights on numerical error in symplectic integration

TL;DR

This paper introduces a new family of symplectic integrators derived from a novel method, demonstrating their ability to reduce energy errors in perturbed Hamiltonian systems through parameter tuning.

Contribution

It presents a new class of symplectic integrators based on special symplectic manifolds, with tunable parameters to improve energy conservation in numerical simulations.

Findings

Energy error can be significantly reduced with tuned integrators.

The new integrators outperform classical schemes in perturbed systems.

Geometrical explanation supports the observed error reduction.

Abstract

We implement and investigate the numerical properties of a new family of integrators connecting both variants of the symplectic Euler schemes, and including an alternative to the classical symplectic mid-point scheme, with some additional terms. This family is derived from a new method, introduced in a previous study, for generating symplectic integrators based on the concept of special symplectic manifold. The use of symplectic rotations and a particular type of projection keeps the whole procedure within the symplectic framework. We show that it is possible to define a set of parameters that control the additional terms providing a way of "tuning" these new symplectic schemes. We test the "tuned" symplectic integrators with the perturbed pendulum and we compare its behavior with an explicit scheme for perturbed systems. Remarkably, for the given examples, the error in the energy integral can be reduced considerably. There is a natural geometrical explanation, sketched at the end of this paper. This is the subject of a parallel article where a finer analysis is performed. Numerical results obtained in this paper open a new point of view on symplectic integrators and Hamiltonian error.