Vol 3, No 2 (2012)

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Observed errors in numerically preserving the Hamiltonian of the simple pendulum by multivalue integrators Geometric numerical integration of Hamiltonian problems is devoted to accurately and efficiently maintaining along the numerical solution the invariants preserved along the exact one. Runge-Kutta methods exhibit some fundamental conservation properties if they are symplectic. Even if multivalue methods cannot be symplectic, it is possible to enforce a nearly canonical behavior: this is made possible if the numerical method generates bounded parasitic components over long intervals. This aspect is discussed in R. D’Ambrosio, G. De Martino, B. Paternoster, CAIM Vol 3, No 2, 2012 doi: 10.1685/journal.caim.412 The figures, reported in the mentioned paper, show the errors observed in conserving the Hamiltonian of the simple pendulum by three different multivalue integrators: the top picture regards a method whose parasitic components are non-bounded, while the others are originated by applying two new multivalue methods with bounded parasitic components, both derived in the above paper. The numerical evidence confirms that, if parasitism is controlled, the Hamiltonian of the mechanical system is accurately preserved over long time intervals.

Image Credit: Raffaele D'Ambrosio, Giuseppe De Martino, Beatrice Paternoster, Dipartimento di Matematica, Facoltà di Scienze MM.FF.NN., Università degli Studi di Salerno, Italy

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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909