Overview

We now turn to spectral differentiation on a bounded, periodic grid. This process was stated in the form of an N N matrix operation in equation (1.5). Whereas the last chapter dealt with the infinite matrix (1.4) corresponding to the unbounded grid , this chapter develops a practical scheme for computation. Mathematically, there are close connections between the two schemes, and the derivation of our spectral method will follow the same pattern as before. The difference is that the semidiscrete Fourier transform is replaced by the DFT (discrete Fourier transform), which can be computed by the FFT (Fast Fourier Transform) [BrHe95].

At first sight, the requirement of periodicity may suggest that this method has limited relevance for practical problems. Yet periodic grids are surprisingly useful in practice. Often in scientific computing a phenomenon is of interest that is unrelated to boundaries, such as the interaction of solitons in the Korteweg-de Vries (KdV) equation (see p. 112) or the behavior of homogeneous turbulence (a classic reference is [OrPa72]). For such problems, periodic boundary conditions often prove the best choice for computation. In addition, some geometries are physically periodic, such as crystal lattices or rows of turbine blades. Finally, even if the physics is not periodic, the co-ordinate space may be, as is the case for a ? or variable in a computation involving polar or spherical coordinates (see chapter 11). Indeed, among the most common of all spectral methods are methods that mix periodic grids...