High-order FD and PS methods are particularly advantageous in cases of

• high smoothness of solution (but note again the discussion in Section 4.2),

• stringent error requirement,

• long time integrations, and

• more than one space dimension.

Because the PS methods for periodic and nonperiodic problems are quite different, the two cases are discussed separately in what follows. In both cases, we find that the PS methods compare very favorably against FD methods in simple model situations. However, in cases with complex geometries or severe irregularities in the solutions, lower-order FD (or FE) methods may be both more economical and more robust.

Especially for nonperiodic problems, it can be difficult to estimate a priori the computational expense required to solve a problem to a desired accuracy. Many implementation variations are possible, and the optimal selection of formal orders of accuracy, level of grid non-uniformity, and so forth may well turn out to depend not only on the problem type, but also on the solution regimes that are studied. Therefore, it makes sense to keep open as many of these implementation options as possible while developing application codes. One technique is to write an FD code of variable order of accuracy on a grid with variable density (using the algorithm in Section 3.1 and Appendix C). By simply changing parameter values, one can then explore (and exploit) the full range of methods from low-order FD on a uniform grid to Chebyshev (Legendre, etc.) and other PS methods. Obviously, it is also desirable to structure codes so that time stepping methods (if present) are easily interchangeable.