Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods

TL;DR

This paper develops continuous-stage numerical methods that preserve energy in non-canonical Hamiltonian systems, ensuring invariants like energy and Casimirs are maintained, with proven theoretical conditions and demonstrated effectiveness.

Contribution

It introduces algebraic conditions for energy preservation in continuous-stage methods and constructs a new symmetric integrator of order 2m for non-canonical Hamiltonian systems.

Findings

New energy-preserving integrators constructed and analyzed.

Theoretical conditions for energy, symmetry, and Casimir preservation established.

Numerical experiments confirm the methods' effectiveness.

Abstract

As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper, we are concerned with the energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms of the Butcher coefficients for ensuring the energy preservation, symmetry and quadratic-Casimir preservation respectively are presented. With the presented condition and in use of orthogonal expansion techniques, the construction of energy-preserving integrators is examined. A new class of energy-preserving integrators which is symmetric and of order $2m$ is constructed. Some numerical results are reported to verify our theoretical analysis and show the effectiveness of our new methods.