In this section the basic concept of spectral methods in one spatial dimension is presented. Some general properties of the approximation of functions by polynomials are introduced. The main formulae of the spectral expansion are then given and two sets of polynomials are discussed (Legendre and Chebyshev polynomials). A particular emphasis is put on convergence properties (i.e., the way the spectral approximation converges to the real function).
In Section 2.5, three different methods of solving an ordinary differential equation (ODE) are exhibited and applied to a simple problem. Section 2.6 is concerned with multidomain techniques. After giving some motivations for the use of multidomain decomposition, four different implementations are discussed, as well as their respective merits. One simple example is given, which uses only two domains.
For problems in more than one dimension see Section 3.
Living Rev. Relativity 12, (2009), 1
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