Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Abstract

We study the implicit time discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization that reduces the numerical dispersion while allowing the use of a large time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating modes, the classical θ-scheme is studied. We present complete new proofs of stability using energy-based approaches that provide uniform results with respect to the featured time step. A dispersion analysis confirms that θ=1/12 reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new θ-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs are also provided for this new scheme. Theoretical results are illustrated with numerical experiments corresponding to the simulation of a realistic piano string.