Spectral methods in physics

1.2 Spectral methods in physics

We do not give here all the fields of physics in which spectral methods are employed, but sketching the variety of equations and physical models that have been simulated with such techniques. Spectral methods originally appeared in numerical fluid dynamics, where large spectral hydrodynamic codes have been regularly used to study turbulence and transition to the turbulence since the seventies. For fully resolved, direct numerical calculations of Navier–Stokes equations, spectral methods were often preferred for their high accuracy. Historically, they also allowed for two or three-dimensional simulations of fluid flows, because of their reasonable computer memory requirements. Many applications of spectral methods in fluid dynamics have been discussed by Canuto et al. [56

, 58

], and the techniques developed in that field are of some interest to numerical relativity.

From pure fluid-dynamics simulations, spectral methods have rapidly been used in connected fields of research: geophysics [190 ], meteorology and climate modeling [217 ]. In this last research category, global circulation models are used as boundary conditions to more specific (lower-scale) models, with improved micro-physics. In this way, spectral methods are only a part of the global numerical model, combined with other techniques to bring the highest accuracy, for a given computational power. A solution to the Maxwell equations can, of course, also be obtained with spectral methods and therefore, magneto-hydrodynamics (MHD) have been studied with these techniques (see, e.g., Hollerbach [119 ]). This has been the case in astrophysics too, where, for example, spectral three-dimensional numerical models of solar magnetic dynamo action realized by turbulent convection have been computed [52 ]. And Kompaneet’s equation, describing the evolution of photon distribution function in a plasma bath at thermal equilibrium within the Fokker-Planck approximation, has been solved using spectral methods to model the X-ray emission of Her X-1 [33 , 40 ]. In simulations of cosmological structure formation or galaxy evolution, many N-body codes rely on a spectral solver for the computation of the gravitational force by the particle-mesh algorithm. The mass corresponding to each particle is decomposed onto neighboring grid points, thus defining a density field. The Poisson equation giving the Newtonian gravitational potential is then usually solved in Fourier space for both fields [118 ].

To our knowledge, the first published result of the numerical solution of Einstein’s equations, using spectral methods, is the spherically-symmetric collapse of a neutron star to a black hole by Gourgoulhon in 1991 [95 ]. He used spectral methods as they were developed in the Meudon group by Bonazzola and Marck [44 ]. Later studies of quickly-rotating neutron stars [41 ] (stationary axisymmetric models), the collapse of a neutron star in tensor-scalar theory of gravity [157 ] (spherically-symmetric dynamic spacetime), and quasiequilibrium configurations of neutron star binaries [39 ] and of black holes [110 ] (three-dimensional and stationary spacetimes) have grown in complexity, up to the three-dimensional time-dependent numerical solution of Einstein’s equations [37 ]. On the other hand, the first fully three-dimensional evolution of the whole Einstein system was achieved in 2001 by Kidder et al. [127 ], where a single black hole was evolved to $t ≃ 600M – 1300M$ using excision techniques. They used spectral methods as developed in the Cornell/Caltech group by Kidder et al. [125 ] and Pfeiffer et al. [172 ]. Since then, they have focused on the evolution of black-hole–binary systems, which has recently been simulated up to merger and ring down by Scheel et al. [186 ]. Other groups (for instance Ansorg et al. [10 ], Bartnik and Norton [21 ], Frauendiener [81 ] and Tichy [219 ]) have also used spectral methods to solve Einstein’s equations; Sections 5 and 6 are devoted to a more detailed review of these works.