The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
This work provides a novel theoretical framework for energy-preserving numerical integration in Hamiltonian systems, offering a finite-step alternative to classical symplectic methods.
The paper addresses the challenge of energy conservation in numerical integration of Hamiltonian systems, proposing a sequence of methods that become energy-preserving after a finite number of steps, overcoming the limitation of symplectic methods that require infinite processes.
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.