The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
arXiv:1010.4538 · doi:10.1016/j.amc.2011.03.022
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.
15 pages
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- Energy conservation issues in the numerical solution of the semilinear wave equation
- On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems
- Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
- Energy-conserving methods for Hamiltonian Boundary Value Problems and applications in astrodynamics
- Line Integral solution of Hamiltonian PDEs
- A note on the continuous-stage Runge-Kutta-(Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)
- Arbitrarily high-order energy-conserving methods for Poisson problems
- Efficient implementation of Radau collocation methods