Enhanced HBVMs for the numerical solution of Hamiltonian problems with multiple invariants

TL;DR

This paper introduces Enhanced HBVMs, an improved class of energy-conserving methods for solving Hamiltonian problems with multiple invariants, demonstrating their theoretical properties and effectiveness through numerical tests.

Contribution

The paper develops and analyzes Enhanced HBVMs, extending previous methods to better handle Hamiltonian problems with multiple invariants.

Findings

Enhanced HBVMs conserve energy and invariants effectively.

Numerical tests confirm improved accuracy and stability.

Methods are theoretically justified and practically validated.

Abstract

Recently, the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), has been proposed for the efficient solution of Hamiltonian problems, as well as for other types of conservative problems. In this paper, we report further advances concerning such methods, resulting in their enhanced version (Enhanced HBVMs, or EHBVMs). The basic theoretical results are sketched, along with a few numerical tests on a Hamiltonian problem, taken from the literature, possessing multiple invariants.