ABSTRACT
We present in this chapter the spectral methods, which are a kind of weighted residuals methods with a specific choice of orthogonal development functions. We start by providing the background principles underlying the three main spectral methods applied to a linear problem. Then, we specialize the presentation to the case of Burgers equation and of Helmholtz-type equations, giving progressively more details in the spectral technique for the case of Fourier and Chebyshev polynomials. The implicit time-discretization of the heat equation is presented exhaustively, so that the practical implementation of the Chebyshev-tau method for this particular case can be done without particular difficulty.
