Energy-conserving finite difference schemes for nonlinear strings

In this paper, we discuss numerical methods for the simulation of nonlinear string vibration. In a musical setting, such vibrations occur under large-amplitude conditions. Though it is simple enough to write down a nonlinear equation for the motion of such a string, the construction of numerical methods, such as finite difference schemes, becomes a much more delicate problem, in particular because of the number of possible discretization approaches, and the lack of spectral analysis tools which are often used to determine stability bounds in the linear case. For this reason, we turn towards more general techniques, which are based around energetic principles, sometimes known collectively as the energy method. Indeed, it is possible to transfer the complete energetic behavior of the nonlinear string to discrete time in such a way that a discrete analogue of the energy is preserved. This leads, in turn, to an energy-based stability guarantee. Such techniques rely in no way on spectral analysis. Various other issues, such as parasitic oscillations, boundary termination, loss modelling and implicit difference schemes are also discussed. Simulations are presented.

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