Spectral Methods

Numerical methods for partial differential equations can be classified into the local and global categories. The finite-difference and finite-element methods are based on local arguments, whereas the spectral method is global in character. In practice, finite-element methods are particularly well suited to problems in complex geome-tries, whereas spectral methods can provide superior accuracy, at the expense of domain flexibility. We emphasize that there are many numerical approaches, such as hp finite-elements and spectral-elements, which combine advantages of both the global and local methods. However in this book, we shall restrict our attentions to the global spectral methods.

The spectral method was introduced in Orszag’s pioneer work on using Fourier series for simulating incompressible flows about four decades ago (cf. Orszag (1971)). The word “spectral” was probably originated from the fact that the Fourier series are the eigenfunctions of the Laplace operator with periodic boundary conditions. This fact and the availability of the fast Fourier transform (FFT) are two major reasons for the extensive applications of Fourier methods to problems with periodic boundary conditions. In practice, a variety of physical problems exhibit periodicity. For instance, some problems are geometrically and physically periodic, such as crystal structures and homogeneous turbulence. On the other hand, many problems of scientific interest, such as the interaction of solitary waves and homogeneous turbulence, can be modeled by PDEs with periodic boundary conditions. Furthermore, even if an original problem is not periodic, the periodicity may be induced by using coordinate transforms, such as polar, spherical and cylindrical coordinates. Indeed, there are numerous circumstances where the problems are periodic in one or two directions, and non-periodic in other directions. In such cases, it is natural to use Fourier series in the periodic directions and other types of spectral expansions, such as Legendre or Chebyshev polynomials, in the non-periodic directions (cf. Chap. 7).

The Fourier spectral method is only appropriate for problems with periodic boundary conditions. If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, and reduces the global convergence rate to O(N^-1) (cf. Gottlieb and Orszag (1977)). Consequently, one should not apply a Fourier method to problems with non-periodic boundary conditions. Instead, one should use orthogonal polynomials which are eigenfunctions of some singular Sturm-Liouville problems. The commonly used orthogonal polynomials include the Legendre, Chebyshev, Hermite and Laguerre polynomials.

We consider in this chapter spectral algorithms for solving the two-point boundary value problem. \( -\varepsilon \rm{U}^{n}+ p(x){\rm{U}=F}, in 1:=(-1,1),\)

This chapter is devoted to spectral approximations of the Volterra integral equation (VIE): \(y(t)+\int_{o}^{t}R(t,\tau)y(\tau)=f(t),\,\,t\epsilon[0,T],\)

High-order differential equations often arise from mathematical modeling of a variety of physical phenomena. For example, higher even-order differential equations may appear in astrophysics, structural mechanics and geophysics, and higher odd-order differential equations, such as the Korteweg-de Vries (KdV) equation, are routinely used in modeling nonlinear waves and nonlinear optics.

We study in this chapter spectral approximations by orthogonal polynomials/functions on unbounded intervals, such as Laguerre and Hermite polynomials /functions and rational functions. Considerable progress has been made in the last two decades in using these orthogonal systems for solving PDEs in unbounded domains (cf. Chap. 17 in Boyd (2001) and a more recent review article Shen and Wang (2009)).

The main goals of this chapter are (a) to design efficient spectral algorithms for solving second-order elliptic equations in separable geometries; and (b) to provide a basic framework for error analysis of multi-dimensional spectral methods.

We consider in this chapter several multi-dimensional problems, which (a) are of current interest; (b) are suitable for spectral approximations; and (c) can be efficiently solved by using the basic spectral algorithms developed in previous chapters. These include steady state problems: the Helmholtz equation for acoustic scattering and the Stokes equations, as well as time-dependent problems including the Allen-Cahn equation, the Cahn–Hilliard equation, the Navier–Stokes equations, and the Gross–Pitaevskii equation. For applications of spectral methods to other multidimensional problems in science and engineering, we refer, for instance, to Boyd (2001), Canuto et al. (2006), Hesthaven et al. (2007) and the references therein.