Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems

TL;DR

This paper introduces Enhanced Line Integral Methods (ELIMs), an extension of energy-conserving Runge-Kutta methods, specifically designed to conserve multiple invariants in Hamiltonian systems, with theoretical analysis and numerical validation.

Contribution

It presents ELIMs, a novel class of methods that extend LIMs to conserve multiple invariants in Hamiltonian problems, improving upon existing energy-conserving techniques.

Findings

ELIMs successfully conserve multiple invariants in Hamiltonian systems.

Numerical tests confirm the theoretical properties of ELIMs.

The methods outperform some existing approaches in preserving invariants.

Abstract

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.