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.

2.1 Best polynomial approximation

2.2 Interpolation on a grid

2.3 Polynomial interpolation

2.3.1 Orthogonal polynomials

2.3.2 Gaussian quadratures

2.3.3 Spectral interpolation

2.3.4 Two equivalent descriptions

2.4 Usual polynomials

2.4.1 Sturm–Liouville problems and convergence

2.4.2 Legendre polynomials

2.4.3 Chebyshev polynomials

2.4.4 Convergence properties

2.4.5 Trigonometric functions

2.4.6 Choice of basis

2.5 Spectral methods for ODEs

2.5.1 The weighted residual method

2.5.2 The tau method

2.5.3 The collocation method

2.5.4 Galerkin method

2.5.5 Optimal methods

2.6 Multidomain techniques for ODEs

2.6.1 Motivations and setting

2.6.2 The multidomain tau method

2.6.3 Multidomain collocation method

2.6.4 Method based on homogeneous solutions

2.6.5 Variational method

2.6.6 Merits of the various methods

2.2 Interpolation on a grid

2.3 Polynomial interpolation

2.3.1 Orthogonal polynomials

2.3.2 Gaussian quadratures

2.3.3 Spectral interpolation

2.3.4 Two equivalent descriptions

2.4 Usual polynomials

2.4.1 Sturm–Liouville problems and convergence

2.4.2 Legendre polynomials

2.4.3 Chebyshev polynomials

2.4.4 Convergence properties

2.4.5 Trigonometric functions

2.4.6 Choice of basis

2.5 Spectral methods for ODEs

2.5.1 The weighted residual method

2.5.2 The tau method

2.5.3 The collocation method

2.5.4 Galerkin method

2.5.5 Optimal methods

2.6 Multidomain techniques for ODEs

2.6.1 Motivations and setting

2.6.2 The multidomain tau method

2.6.3 Multidomain collocation method

2.6.4 Method based on homogeneous solutions

2.6.5 Variational method

2.6.6 Merits of the various methods

Living Rev. Relativity 12, (2009), 1
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