On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae

TL;DR

This paper introduces a family of symplectic integrators based on Gauss collocation formulas, which can be tuned to conserve energy exactly while maintaining high order accuracy.

Contribution

It presents a new parameterized family of symplectic integrators that can be adjusted to preserve energy exactly, extending classical Gauss methods.

Findings

The integrators are symplectic and can be energy-preserving.

Energy conservation is achievable through parameter tuning at each step.

The methods retain high order accuracy comparable to Gauss collocation formulas.

Abstract

We introduce a new family of symplectic integrators depending on a real parameter. When the paramer is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null value of the parameter, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as an order 2s-2 (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the free parameter may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting symplectic, energy conserving method shares the same order 2s as the generating Gauss formula.