Conservative numerical methods for sound synthesis

Abstract

Direct numerical simulation techniques, such as finite difference and finite element methods, are playing an increasingly important role in sound synthesis based on physical models of musical instruments. While they can lead to enormous improvements in sound quality over existing methods, a problem of mounting importance is that of maintaining numerical stability when the system to be simulated is strongly nonlinear, as is the case for many musical instruments. This problem is especially acute if real‐time synthesis is to be an eventual goal. Standard stability checking machinery, such as frequency domain (von Neumann) analysis, does not immediately apply, and, often, the most straightforward numerical methods may exhibit such instabilities in an unpredictable manner. In this paper, various techniques for the construction and analysis of numerical methods for strongly nonlinear musical systems are presented; they are based on the translation of the principle of conservation or dissipation of energy to discrete time and can lead directly to sufficient nonlinear stability conditions, without invoking frequency domain concepts. Special attention is paid to the nonlinear string, bar, and plate models. Numerical results and sound examples will be presented.