The connection between pseudospectral and finite difference methods is investigated. A key theme of this book is to exploit this connection and to obtain powerful and flexible spectral methods. Finite difference schemes are local whereas spectral schemes are global. Usually the approximation is performed by very smooth functions, for example by Chebyshev polynomials or trigonometric functions. For analytical functions, errors typically decay at exponential rate. Further, the method is virtually free of both dissipative and dispersive errors. On the other hand, there are difficulties when using spectral methods: certain boundary conditions, irregular domains, strong shocks.
At present, spectral methods are highly successful in several areas: turbulence, weather prediction, nonlinear waves, seismic modeling. The efficiency of spectral methods benefited greatly from the fast Fourier transform (FFT) algorithm of J. W. Cooley and J. W. Tukey [Math. Comput. 19, 297-301 (1965; Zbl 0127.09002)].
It is shown that pseudospectral methods can be seen as limiting cases of increasing-order finite difference methods. The basic idea goes back to H.-O. Kreiss and J. Oliger [Tellus 24, 199-215 (1972; MR 47.7926)], and was developed by B. Fornberg [SIAM J. Numer. Analysis 12, 509-528 (1975; Zbl 0349.35003) and ibid. 27, No. 4, 904-918 (1990; Zbl 0705.65076)]. In chapter 3 advantages of the finite difference methods are listed. In the remaining chapters, key properties of spectral methods and applications are discussed.