Hamiltonian systems have been extensively studied in the literature since many physical lossless systems can be cast in Hamiltonian form. Numerically, various approaches exist, particularly regarding geometric integrators. Among these, energy-preserving schemes represent an important class, since it is known that energy conservation can be related to the stability of the time-stepping routine when the potential energy is bounded from below. However, classic energy-preserving integrators are most often implicit in nature, requiring the solution of nonlinear algebraic systems at each time step. Explicit integrators, on the other hand, fail to preserve the numerical energy in most cases of interest and lead to unstable simulations. In this talk, I am going to present a new class of schemes for the integration of Hamiltonian systems with non-negative potential energy. These schemes are based on the idea of quadratisation and, under some restrictions on the form of the mass matrix, are both energy-conserving and explicit. Cases of interest in acoustics (i.e., the von Karman plate and the geometrically exact nonlinear string) will be studied in detail, showing that compute times for the new schemes are on par with classic explicit integrators such as Stormer-Verlet, while remaining stable even at large amplitudes of vibration.
