The modeling of time-dependent physical systems is traditionally the ultimate goal in numerical simulations. Within the field of numerical relativity, the need for studies of dynamic systems is even more pronounced because of the search for gravitational wave patterns. Unfortunately, as presented in Section 4.1, there is no efficient spectral time discretization yet and one normally uses finite-order time-differentiation schemes. Therefore, in order to get high temporal accuracy, one must use high-order explicit time-marching schemes (e.g., fourth or sixth-order Runge–Kutta [49]). This requires quite a lot of computational power and might explain why, except for gravitational collapse [95, 156], very few studies using spectral methods have dealt with dynamic situations until the Caltech/Cornell group began to use spectral methods in numerical relativity in the early part of this century [128, 127]. This group now has a very well-developed pseudospectral collocation code, “Spectral Einstein Code” (SpEC), for the solution of full three-dimensional dynamic Einstein equations.

In this section, we review the status of numerical simulations that use spectral methods in some fields of general relativity and relativistic astrophysics. Although we may give at the beginning of each section a very short introduction to the context of the relevant numerical simulations, our point is not to give detailed descriptions of them, as dedicated reviews exist for most of the themes presented here and the interested reader should consult them for details of the physics and comparisons with other numerical and analytic techniques. Among the systems that have been studied, one can find gravitational collapse [84] (supernova core collapse or collapse of a neutron star to a black hole), oscillations of relativistic stars [205, 130] and evolution of “vacuum” spacetimes. These include the cases of pure gravitational waves or scalar fields, evolving in the vicinity of a black hole or as (self-gravitating) perturbations of Minkowski flat spacetime. Finally, we will discuss the situation of compact binary [174, 31] spectral numerical simulations.

6.1 Single Stars

6.1.1 Supernova core collapse

6.1.2 Collapse to a black hole

6.1.3 Relativistic stellar pulsations

6.2 Vacuum and black hole evolutions

6.2.1 Formulation and boundary conditions

6.2.2 Gauges and wave evolution

6.2.3 Black hole spacetimes

6.3 Binary systems

6.3.1 Neutron star binaries

6.3.2 Black-hole–neutron-star binaries

6.3.3 Black hole binaries

6.1.1 Supernova core collapse

6.1.2 Collapse to a black hole

6.1.3 Relativistic stellar pulsations

6.2 Vacuum and black hole evolutions

6.2.1 Formulation and boundary conditions

6.2.2 Gauges and wave evolution

6.2.3 Black hole spacetimes

6.3 Binary systems

6.3.1 Neutron star binaries

6.3.2 Black-hole–neutron-star binaries

6.3.3 Black hole binaries

Living Rev. Relativity 12, (2009), 1
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