Grid Functions and Finite Difference Operators

In this short chapter, the basic operations used in the construction finite difference schemes for distributed problems are introduced. The treatment will be somewhat abbreviated due partly to the fact that much of this material appears elsewhere, and partly because many of the underlying ideas were introduced in Chapter 2. As was the case for this earlier chapter, the presentation will again encompass both frequency domain analysis, and pure time-space (energy) techniques. The full power of energy methods will become evident here, not just because of its capability for arriving at stability conditions even for strongly nonlinear problems (some important musical examples of which will appear in subsequent chapters), but because of the ease with which appropriate numerical boundary conditions may be extracted; the proper setting of boundary conditions is always problematic, and is not at all well covered even in the best finite difference texts. This is not to say that frequency domain techniques are not useful; as was mentioned earlier, they are able to yield much important information regarding numerical dispersion, which, in musical simulations, leads to perceptually audible deviations from the solution to a model system.

Grid functions and difference operators in a single spatial dimension are introduced first, in §5.1, followed by a discussion of frequency domain analysis, the matrix interpretation of difference operators, and finally an introduction to inner product definitions and various manipulations useful in energetic analysis. A similar treatment is provided for grid functions and finite difference operators in two spatial dimensions, first in the case of Cartesian coordinates in §5.2, and then in polar coordinates, which are of special interest in many musical applications, in §5.3.

References for the chapter include: [210,102,90,2,243,200,79,226,89]