One of the most important drawbacks to the use of time domain methods relates to the effect of numerical dispersion, which leads, at least in the linear case, to mistuning of modal frequencies. Numerical dispersion itself results from insufficient accuracy in a numerical method. While the perceptual importance of numerical dispersion in musical sound synthesis applications, which is generally limited to the upper end of the human hearing range, is a matter of debate, this is a good point to introduce other numerical techniques which have much better accuracy properties.
Finite difference methods and finite element methods are based on essentially local discretizations of a model system; generally, one models the various partial derivatives at a given grid point or node through combinations of values at neighboring points. As was mentioned early on, better accuracy may be achieved as more neighboring points are used in the calculation; this renders the calculations less local. In the most well-known manifestations of spectral methods13.1 [41,225,90,166] approximations are global; a derivative calculated at a given grid point will make use of information from the entire spatial domain of the problem. In fact, even the concept of a grid function is not entirely useful in this setting, as the solution itself is considered to be decomposed into an entire set of basis functions--modal synthesis[1], particularly in the form discovered independently by Garber[93], is an example of a very simple spectral method based on a Fourier decomposition.
There are many different forms of spectral methods, dependent chiefly of the types of expansions used to represent the solution (often Fourier or polynomial series, discussed in §13.1), and on the constraints one places on the solution: if the error is to be minimized at a finite collection of points in the spatial domain, one speaks of a collocation method (see §13.2), and if an error with respect to the expansion basis used to represent the solution is minimized, then a Galerkin method results (see §13.3). More general weighted residual methods have been developed, but in this short chapter, only these two forms will be discussed, particularly with reference to their suitability for musical sound synthesis. Spectrally accurate derivative approximations, as well as the use of non-uniform Chebyshev grids, and fast evaluation through the use of the fast Fourier transform are discussed in §13.4. As spectrally accurate approximations are generally applied to spatial derivatives, global accuracy may be achieved through the use of standard high-order time differencing strategies--these are briefly summarized in §13.5. Finally, musical applications to the stiff string and linear plate problems are presented, in §13.6 and 13.7, respectively.
References for this chapter include [225,90,41,,166,128,127,1,93,98]