Energy and quadratic invariants preserving integrators of Gaussian type

TL;DR

This paper introduces a new family of symplectic integrators that preserve energy and quadratic invariants for Hamiltonian problems, generalizing Gauss collocation methods with tunable parameters.

Contribution

A novel class of energy and quadratic invariants preserving integrators based on a parameterized family of symplectic methods, extending classical Gauss collocation formulas.

Findings

Methods can be tuned to conserve energy at each step.

Preserves both energy and quadratic invariants.

Maintains order 2s for s stages.

Abstract

Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems. In the mainstream of this research, we have defined a new family of symplectic integrators depending on a real parameter. When it 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 a symplectic perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter a may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula, and is able to preserve both energy and quadratic invariants.