Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

This paper considers a general class of nonlinear systems, “nonlinear Hamiltonian systems of wave equations”. The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). 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 trivial. 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. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the “geometrically exact model”, or approximations of this model. Numerical results are presented.