Up to this point, we have described basic PS implementations. However, many variations are possible, offering advantages in different respects. In this chapter, we discuss a few of these variations.

Use of additional information from the governing equations

This idea (like most others) is best described through the use of examples. It requires that the problem be manipulated analytically (e.g., by repeated differentiation) to provide more information than is immediately available from its original formulation.

Example 1. Exploit additional derivative information at the boundaries when solving the eigenvalue problem uxx = λu, u(±1) =0.

Since u(±1) = 0, clearly also u″(±1) = 0 and u″″(±1) = 0 (we label these as “extra” boundary conditions – for this example, we ignore that this pattern continues indefinitely and that u becomes periodic). The boundary information on u″ and u″″ can be exploited in different ways.

A, Reduce the largest spurious EVs (cf. Figure 4.4-2). To each extra boundary condition (such as u″(−1) = 0) corresponds a one-sided difference stencil. From each row of the DM, like those shown in Figures 4.3-1(c) and (d), we can subtract any multiples of these stencils without compromising the spectral accuracy. The multiples can be chosen to minimize the sum of the squares of the elements of the resulting DM.