Energy Preserving Schemes for Nonlinear Hamiltonian Systems of Wave Equations. Application to the Vibrating Piano String.

Juliette Chabassier 1 Patrick Joly 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : The problem of the vibration of a string is well known in its linear form, describing the transversal motion of a string, nevertheless this description does not explain all the observations well enough. Nonlinear coupling between longitudinal and transversal modes seams to better model the piano string, as does for instance the ''geometrically exact model'' (GEM). This report introduces a general class of nonlinear systems, ''nonlinear hamiltonian systems of wave equations'', in which fits the GEM. Mathematical study of these systems is lead in a first part, showing central properties (energy preservation, existence and unicity of a global smooth solution, finite propagation velocity \ldots). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of ''preserving schemes'' is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class.
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[Research Report] RR-7168, INRIA. 2010, pp.70
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Juliette Chabassier, Patrick Joly. Energy Preserving Schemes for Nonlinear Hamiltonian Systems of Wave Equations. Application to the Vibrating Piano String.. [Research Report] RR-7168, INRIA. 2010, pp.70. 〈inria-00444470〉

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