About spectral methods

1.1 About spectral methods

When doing simulations and solving PDEs, one faces the problem of representing and deriving functions on a computer, which deals only with (finite) integers. Let us take a simple example of a function $f : [− 1,1] → ℝ$ . The most straightforward way to approximate its derivative is through finite-difference methods: first one must setup a grid

{xi}i=0...N ⊂ [− 1,1 ]

of $N + 1$ points in the interval, and represent $f$ by its $N + 1$ values on these grid points

{fi = f (xi)}i=0...N .

Then, the (approximate) representation of the derivative $f′$ shall be, for instance,

If we suppose an equidistant grid, so that $∀i < N, xi+1 − xi = Δx = 1∕N$ , the error in the approximation (1

) will decay as $Δx$ (first-order scheme). One can imagine higher-order schemes, with more points involved for the computation of each derivative and, for a scheme of order $n$ , the accuracy can vary as $n (Δx ) = 1∕N n$ .

Spectral methods represent an alternate way: the function $f$ is no longer represented through its values on a finite number of grid points, but using its coefficients (coordinates) ${ci} i=0...N$ in a finite basis of known functions ${Φi}i=0...N$

A relatively simple case is, for instance, when $f(x)$ is a periodic function of period two, and the $Φ2i(x) = cos(πix ),Φ2i+1 (x ) = sin(πix )$ are trigonometric functions. Equation (2

) is then nothing but the truncated Fourier decomposition of $f$ . In general, derivatives can be computed from the $ci$ ’s, with the knowledge of the expression for each derivative $Φ ′i(x)$ as a function of ${Φi}i=0...N$ . The decomposition (2

) is approximate in the sense that ${Φi } i=0...N$ represent a complete basis of some finite-dimensional functional space, whereas $f$ usually belongs to some other infinite-dimensional space. Moreover, the coefficients $ci$ are computed with finite accuracy. Among the major advantages of using spectral methods is the rapid decay of the error (faster than any power of $1∕N$ , and in practice often exponential $e− N$ ), for well-behaved functions (see Section 2.4.4); one, therefore, has an infinite-order scheme.

In a more formal and mathematical way, it is useful to work within the methods of weighted residuals (MWR, see also Section 2.5). Let us consider the PDE

where $L$ is a linear operator, $B$ the operator defining the boundary conditions and $s$ is a source term. A function $¯u$ is said to be a numerical solution of this PDE if it satisfies the boundary conditions (4

) and makes “small” the residual

If the solution is searched for in a finite-dimensional subspace of some given Hilbert space (any relevant $L2 U$ space) in terms of the expansion (2

), then the functions ${Φ (x)} i i=0...N$ are called trial functions and, in addition, the choice of a set of test functions ${ξi(x)}i=0...N$ defines the notion of smallness for the residual by means of the Hilbert space scalar product

Within this framework, various numerical methods can be classified according to the choice of the trial functions:

Finite differences: the trial functions are overlapping local polynomials of fixed order (lower than $N$ ).
Finite elements: the trial functions are local smooth functions, which are nonzero, only on subdomains of $U$ .
Spectral methods: the trial functions are global smooth functions on $U$ .

Various choices of the test functions define different types of spectral methods, as detailed in Section 2.5. Usual choices for the trial functions are (truncated) Fourier series, spherical harmonics or orthogonal families of polynomials.