The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity

TL;DR

This paper explores the role of infinite and infinitesimal concepts in numerical methods for Hamiltonian systems, proposing a sequence of methods that preserve energy after finitely many steps, bridging continuous and discrete symplecticity.

Contribution

It introduces a new sequence of numerical methods that achieve energy preservation after finite steps, extending the classical symplectic approach to a discrete setting.

Findings

Sequence of methods shares the same spectrum as classical methods

Energy preservation occurs after finite steps in the sequence

Bridges the gap between continuous infinitesimal transformations and discrete methods

Abstract

When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, among which the Hamiltonian function itself, assumes a central role. The classical approach to this problem has led to the definition of symplectic methods, among which we mention Gauss-Legendre collocation formulae. Indeed, in the continuous setting, energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations. However, this infinite process cannot be directly transferred to the discrete setting. By following a different approach, in this paper we describe a sequence of methods, sharing the same essential spectrum (and, then, the same essential properties), which are energy preserving starting from a certain element of the sequence on, i.e., after a finite number of steps.