9 Spatial Approximations: Spectral Methods

In this section, we review some of the theory for spectral spatial approximations, and their applications in numerical relativity. These are global representations, which display very fast convergence for smooth enough functions. They are therefore very well suited for Einstein’s vacuum equations, where physical shocks are not expected since they can be written in linearly-degenerate form, as discussed at the beginning of Section 3.3.

We start in Section 9.1 discussing expansions onto orthogonal polynomials, which are solutions to Sturm–Liouville problems. In those cases it is easy to see that for smooth functions the decay of the error in truncated expansions with respect to the number of polynomials is in general faster than any power law, which is usually referred to as spectral convergence. Next, in Section 9.2 we discuss a few properties of general orthogonal polynomials; most important that they can be generated through a three-term recurrence formula. Follows Section 9.3 with a discussion of the most-used families of polynomials in bounded domains; namely, Legendre and Chebyshev ones, including the minmax property of Chebyshev points. Approximating integrals through a global interpolation with a careful choice of nodal points makes it possible to maximize the degree with respect to which they are exact for polynomials (Gauss quadratures). When applied to compute discrete truncated expansions, they lead to two remarkable features. One of them is SBP for Legendre polynomials, in analogy with the FD version discussed in Section 8.3. As in that case, SBP can also be sometimes used to show semi-discrete stability when solving time-dependent partial differential equations (PDEs). The second one is an exact equivalence, for general Jacobi polynomials, between the resulting discrete expansions and interpolation at the Gauss points, a very useful property for collocation methods. Gauss quadratures and SBP are discussed in Section 9.4, followed by interpolation at Gauss points in Section 9.5. In Sections 9.6, 9.7 and 9.8 we discuss spectral differentiation, the collocation method for time-dependent PDEs, and applications to numerical relativity.

The results for orthogonal polynomials to be discussed are classical ones, but we present them because spectral methods are less widespread in the relativity community, at least compared to FDs. The proofs and a detailed discussion of many other properties can be found in, for example, [197] and references therein. [237] is a modern approach to the use of spectral methods in time-dependent problems with the latest developments, while [70] discusses many issues, which appear in applications, and [167] presents a very clear practical guide to spectral methods, in particular to the collocation approach. A good fraction of our presentation of this section follows [197] and [237], to which we refer when we do not provide any other references, or for further material.