Overview

Spectral computations are frequently carried out in multidimensional domains in which one has different kinds of boundary conditions in the different dimensions. One of the most common examples is the use of polar coordinates in the unit disk,

Including a third variable z or would bring us to cylindrical or spherical coordinates.

The most common way to discretize the disk spectrally is to take a periodic Fourier grid in ? and a nonperiodic Chebyshev grid in r:

Specifically, the grid in the r direction is transformed from the usual Chebyshev grid for x ? [ ?1, 1] by r = (x+ 1)/2. The result is a polar grid that is highly clustered near both the boundary and the origin, as illustrated in Figure 11.1. Grids like this are convenient and commonly used, but they have some drawbacks. One dificulty is that while it is sometimes advantageous to have points clustered near the boundary, it may be wasteful and is certainly inelegantto devote extra grid points to the very small region near the origin, if the solution is smooth there. Another is that for time-dependent problems, these small cells near the origin may force one to use excessively small time steps for numerical stability. Accordingly, various authors have found alternative ways to treat the region near r = 0. We shall describe one method of this kind in essentially the formulation proposed by Fornberg [For95, For96, FoMe97]. Closely...