SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Numerical solution of Hamiltonian problems by G-symplectic integrators

B. Paternoster, R. D'Ambrosio and G.D. Martino

Abstract
It is the purpose of this talk to analyze the employ of General Linear Methods (GLMs) for the numerical integration of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, Butcher recently introduced in [1] a concept of near conservation, denoted as G-symplecticity, for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve a very accurate long time conservation of the Hamiltonian. We also focus our attention on the connections between the order of convergence of a GLM and the observable Hamiltonian deviation, by employing the theory of B-series [3]. Moreover, we derive a semi-implicit GLM [2] which results competitive with respect to symplectic Runge-Kutta methods. Numerical results on a selection of Hamiltonian problems are presented, confirming the structure-preserving capability of G-symplectic integrators.

Bibliography
[1] J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 2008.
[2] R. D'Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems, Comm. Appl. Ind. Math, doi: 10.1685/journal.caim.412 (2013).
[3] R. D'Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic integrators, submitted.

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