Chapter 1 Preliminaries 1.1 Some basic ideas of spectral methods 1.2 Orthogonal polynomials 1.3 Chebyshev and Legendre polynomials 1.4 Jacobi polynomials and generalized Jacobi polynomials 1.5 Fast Fourier transform 1.6 Several popular time discretization methods 1.7 Iterative methods and preconditioning 1.8 Error estimates of polynomial approximations Chapter 2 Spectral-Collocation Me