Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points

TL;DR

This paper compares Gauss-Legendre methods and Hamiltonian Boundary Value Methods (BVMs) based on Gauss points, highlighting their energy-preserving properties for polynomial Hamiltonian systems through numerical tests.

Contribution

It introduces Hamiltonian BVMs as a generalization of collocation methods that exactly conserve polynomial Hamiltonians, with numerical comparisons to Gauss-Legendre methods.

Findings

Hamiltonian BVMs precisely conserve polynomial Hamiltonians.

Numerical tests demonstrate the importance of energy conservation.

Hamiltonian BVMs outperform in preserving system behavior.

Abstract

Hamiltonian Boundary Value Methods are a new class of energy preserving one step methods for the solution of polynomial Hamiltonian dynamical systems. They can be thought of as a generalization of collocation methods in that they may be defined by imposing a suitable set of extended collocation conditions. In particular, in the way they are described in this note, they are related to Gauss collocation methods with the difference that they are able to precisely conserve the Hamiltonian function in the case where this is a polynomial of any high degree in the momenta and in the generalized coordinates. A description of these new formulas is followed by a few test problems showing how, in many relevant situations, the precise conservation of the Hamiltonian is crucial to simulate on a computer the correct behavior of the theoretical solutions.